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Stationary action and Lagrangian dynamics

1 · Prerequisites

2 · Summary

For a nonempty configuration domain, this page is the variational companion of the force model of cm02-forces-initial-data-and-newtonian-flow and of the constraint branch of cm06-regular-constraints-and-moving-frames. It adds no new physical assumption: given the force branch and a declared potential presentation $(U, d)$ of the force map, The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions defines the kinetic energy $\frac{1}{2} v \cdot M v$ of the labels, the mechanical Lagrangian $L = K - U$ of the presentation, the chart instance of the mathematical action on $[a, b] \subseteq I_{0}$ , the fixed-endpoint first variation and stationarity, with the dimensions of all objects recorded ( $[L] = μ ℓ^{2} τ^{- 2}$ , $[S] = μ ℓ^{2} τ^{- 1}$ ). The first variation of the mechanical action and the Euler-Lagrange expression in model terms computes the first variation in model terms, $δ S (x) [η] = \int (- \nabla_{x} U \cdot η + M x^{'} \cdot η^{'}) d t$ , and identifies the Euler-Lagrange expression with the acceleration residual, $E_{L} (x) = - (M x^{''} + \nabla_{x} U)$ . The stationary-action criterion and the potential-part Euler-Lagrange equation turns this into the equivalence of stationarity, the vanishing of the Euler-Lagrange expression and the model's equation of motion of the potential part, and Stationary curves of the mechanical action are exactly the restrictions of the model's motions (global potential branch) proves the stationary-action principle of the global potential branch: the stationary curves on a closed interval are exactly the restrictions of the model's motions, with the converse passage through the maximal motion supplied by the initial-value and continuation theorems of the force page.

The coordinate and gauge structure is developed by Time-dependent coordinate maps transform the potential-part equation into the substituted Euler-Lagrange equation, which reads a time-dependent coordinate map $X$ with invertible $X_{q}$ into the transport identity $E_{\tilde{L}} (q) = X_{q}^{T} E_{L} (x)$ ; this is a local coordinate change, and it is a global chart only when it is also one-to-one. Thus the full force equation in each such coordinate description retains the residual generalized force; it is the unforced Euler-Lagrange equation of the substituted Lagrangian when $d \equiv 0$ . The result

Total-derivative Lagrangians, the potential representative and scaling: invariant stationary curves with endpoint terms, which records that adding the total time derivative of a $C^{3}$ function on the full time-configuration domain with action dimension changes the action by endpoint values only, that changing the potential representative $U \mapsto U + c (t)$ is an instance of this, and that nonzero dimensionless rescaling of the Lagrangian leaves the stationary curves unchanged. Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces collects the generalized-coordinate dictionary — the immersion $X$ , the substituted Lagrangian with kinetic metric $G = D X^{T} M D X$ , generalized forces $Q = D X^{T} f$ and the declared Lagrange equations with residual generalized force. Full-dimensional nonsingular maps give local coordinates; constraint parametrizations cover the full constrained locus and may have multiple coordinate representatives. The purely mathematical Transport of the Euler-Lagrange expression under a C^3 map: E of the substituted Lagrangian is the Jacobian-transpose multiple of E of the ambient Lagrangian proves the transport identity for non-square immersions. Ideal scleronomic holonomic constraints: the substituted Lagrange equations are equivalent to the constrained model equation then proves the constraint-elimination theorem for the scleronomic ideal branch: on a surjective immersed parametrization $X (Ω) = C \cap Q$ , a coordinate curve solves the Lagrange equations of the substituted Lagrangian exactly when the induced configuration curve is a constrained motion of the model with its unique ideal reaction, so the reaction never appears in the reduced equations. The page closes with Scope of the stationary-action formulation: no minimisation, regular mechanical Lagrangians, no general velocity-constraint equivalence and empirical status: stationarity is not minimisation, the mechanical branch is regular while singular velocity Hessians are excluded, an unrestricted action principle is not generally equivalent to d'Alembert dynamics for velocity constraints, and the whole reformulation is a conditional theorem of the model, not empirical evidence.

Worked checks live on cm07-stationary-action-and-lagrangian-dynamics-examples: the free particle with its exact minimisation identity, the oscillator on a long interval with both signs of the action change, cylindrical local coordinates off the $z$ -axis with kinetic metric $m d i a g (1, r^{2}, 1)$ , a dimensioned total-derivative gauge computation, one normalized nonholonomic model in which unrestricted-action and d'Alembert motions differ, and the planar pendulum recovered through its periodic surjective immersion.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

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The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions

Physical scope

For a nonempty configuration domain, the action functional of the framework's force branch with a declared potential presentation: the kinetic energy of the labels in an inertial frame, the mechanical Lagrangian K - U of the presentation, the open chart and the action integral over C^2 curves with fixed endpoints, the first variation and stationarity, and the dimensions of these objects. If the force postulate is instantiated on an empty configuration domain, there are no curves and this item has no action or stationarity instances. It defines model-derived objects only: it asserts no equation of motion, no conservation law, no minimisation property and no empirical value, and it does not claim that any apparatus is described by a Lagrangian or by the potential branch.

Definition

Adopt one force model of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance: $N \geq 1$ particle labels with fixed masses $m_{1}, \dots, m_{N} > 0$ (clause (i)), an open nondegenerate time window $I_{0} \subseteq R$ , an open configuration domain $Q \subseteq R^{3 N}$ and a force map $F$ with the standing continuity and state-local-Lipschitz hypotheses of clause (iii) of that postulate. Fix one inertial frame with adapted coordinates $(t, x)$ and identify position tuples with points of $R^{3 N} = (R^{3})^{N}$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator); dimensions are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, inner products and norms are Euclidean (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ), and differentiability classes are those of $C^{k}$ Euclidean maps and diffeomorphisms. Let $M$ be the mass operator of clause (d) of Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator, $M (x_{1}, \dots, x_{N}) = (m_{1} x_{1}, \dots, m_{N} x_{N})$ , a symmetric positive definite linear map of $R^{3 N}$ .

(a) Kinetic energy. The kinetic energy of the labels at coordinate velocity $v \in (R^{3})^{N} = R^{3 N}$ is

$K (v) : = \frac{1}{2} \sum_{i = 1}^{N} m_{i} ∣ v_{i} ∣_{2}^{2} = \frac{1}{2} v \cdot M v .$

$K$ is the restriction of a $C^{\infty}$ function of $(R^{3})^{N}$ and has dimension $μ ℓ^{2} τ^{- 2}$ (energy). Its Euclidean gradient is $K^{'} (v) = M v$ , because $\partial K / \partial v_{i} = m_{i} v_{i}$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ).

(b) Mechanical Lagrangian of a potential presentation. Let $(U, d)$ be a potential presentation of the model's force map in the sense of clause (a) of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets: a function $U : I_{0} \times Q \to R$ of class $C^{2}$ with dimension $μ ℓ^{2} τ^{- 2}$ and residual forces $d = (d_{1}, \dots, d_{N})$ such that $F_{i} (t, x, v) = - \nabla_{x_{i}} U (t, x) + d_{i} (t, x, v)$ on the state domain. The mechanical Lagrangian of the presentation is

$L : I_{0} \times Q \times R^{3 N} ⟶ R, L (t, x, v) : = K (v) - U (t, x) .$

$L$ is of class $C^{2}$ on its open domain and has dimension $μ ℓ^{2} τ^{- 2}$ . When $d \equiv 0$ the presentation is a global potential model and $L$ is a Lagrangian of the model itself; when $d \equiv̸ 0$ , $L$ carries the potential part only, and the residual force contribution $d$ is not included in $L$ . The potential representative is part of the declared data: replacing $U$ by $U + c (t)$ for differentiable $c : I_{0} \to R$ replaces $L$ by $L - c (t)$ (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (c)).

(c) Action, variations, stationarity. Let $a < b$ be reals with $[a, b] \subseteq I_{0}$ . If $Q = \emptyset$ , there is no map $x : [a, b] \to Q$ , so this clause has no action, variation or stationarity instance. Assume $Q \neq \emptyset$ for the chart construction below, and put $O : = I_{0} \times Q \times R^{3 N}$ . The set $O$ is open and contains $[a, b] \times Q \times R^{3 N}$ , so with $n : = 3 N$ the pair $(O, L)$ is a time-dependent Lagrangian chart over $[a, b] \times Q$ in the sense of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations. Admissible curves (the $C^{2}$ curves $x : [a, b] \to Q$ with the extension convention of that definition), variations $η$ with $η (a) = η (b) = 0$ , the action

$S (x) : = \int_{a}^{b} L (t, x (t), x^{'} (t)) d t,$

the first variation $δ S (x) [η]$ and stationarity of $x$ are the notions of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations instantiated at this chart; the derivative $x^{'}$ is the coordinate velocity (Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors). The action has dimension $μ ℓ^{2} τ^{- 1}$ (energy multiplied by time). Write $E_{L} (x) : [a, b] \to R^{3 N}$ for the Euler-Lagrange expression of the instantiated chart.

(d) Status. This item defines the action functional and the Lagrangian of a presentation from data already adopted in the model: no new force law, no equation of motion and no measured value are introduced. The first variation of $S$ exists for every admissible curve and variation and its formula, the equivalence of stationarity with the vanishing of $E_{L} (x)$ , and the relation of both to the model's equation of motion are theorems of this page, not part of the definition.

Remarks

The Lagrangian is not unique. The value of $L$ depends on the declared potential representative and on the split of the force map into potential and residual parts. Two presentations of the same force map with different splits produce the same model motions but different Lagrangians; the effect of changing the potential representative is a total time derivative, and the equations of motion are insensitive to it (the boundary-gauge theorem of this page).
Velocity and configuration slots. The three arguments of $L$ are $(t, x, v)$ with $x \in Q$ the configuration and $v$ the coordinate velocity in the fixed inertial frame; the kinetic energy uses the fixed masses and the coordinate velocity, not any generalized velocity. Generalized coordinates enter only through the substitution construction of this page.
Fixed endpoints. Stationarity is defined with variations vanishing at both endpoints, exactly as in A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations. The endpoint-action differential and free-endpoint problems belong to the Hamiltonian development and are not asserted here.

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The first variation of the mechanical action and the Euler-Lagrange expression in model terms

Physical scope

The first variation of the action of a force model with a declared potential presentation, computed in the inertial chart: for every C^2 admissible curve and fixed-endpoint variation the variation exists and equals the integral of minus the potential gradient paired with the variation plus the mass-operator pairing of velocity with the variation's derivative, and the Euler-Lagrange expression of the mechanical Lagrangian is minus the sum of the mass-operator acceleration and the potential gradient. The result is a conditional deduction from the adopted force branch, the declared potential presentation and the cited mathematical first-variation formula; it asserts no equation of motion, which is the subject of the following items.

Statement

Adopt the force model with potential presentation $(U, d)$ and the mechanical Lagrangian $L = K - U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, with masses $m_{1}, \dots, m_{N} > 0$ , configuration domain $Q \subseteq R^{3 N}$ , time window $I_{0}$ , mass operator $M$ and a potential $U : I_{0} \times Q \to R$ of class $C^{2}$ (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator). Let $a < b$ with $[a, b] \subseteq I_{0}$ , let $x : [a, b] \to Q$ be a $C^{2}$ admissible curve and let $η$ be a variation of $x$ (A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations). Then:

$x + ε η$ is an admissible curve for every sufficiently small $∣ ε ∣$ , the function $ε \mapsto S (x + ε η)$ is differentiable at $ε = 0$ , and

$δ S (x) [η] = \int_{a}^{b} [- \nabla_{x} U (t, x (t)) \cdot η (t) + M x^{'} (t) \cdot η^{'} (t)] d t,$

where $\nabla_{x} U = (\nabla_{x_{1}} U, \dots, \nabla_{x_{N}} U)$ is the Euclidean gradient in the configuration slot and the dot is the Euclidean inner product of $R^{3 N}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ );

the Euler-Lagrange expression of $L$ along $x$ is

$E_{L} (x) (t) = - (M x^{''} (t) + \nabla_{x} U (t, x (t))) (t \in [a, b]) .$

The statement is conditional on the adopted force branch and on the declared potential presentation; it selects no equation of motion and asserts no empirical value.

Facts & Assumptions

Given: The model data and presentation of the statement, reals $a < b$ with $[a, b] \subseteq I_{0}$ , a $C^{2}$ curve $x : [a, b] \to Q$ , and a variation $η$ with $η (a) = η (b) = 0$ .

[F1]

The chart instance (The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions): $(O, L)$ with $O = I_{0} \times Q \times R^{3 N}$ is a time-dependent Lagrangian chart over $[a, b] \times Q$ , and admissible curves, variations, action, first variation and Euler-Lagrange expression are those of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations instantiated at it; thus the Euler-Lagrange expression is $E_{L} (x) (t) = L_{x} (t, x (t), x^{'} (t)) - \frac{d}{d t} (L_{v} (t, x (t), x^{'} (t)))$ with $L_{x}$ , $L_{v}$ the Euclidean gradients in the second and third slots (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

[F2]

The first-variation formula (The first variation of a time-dependent action on a finite interval: admissibility, the integral formula and continuity of the Euler-Lagrange coefficient): for the chart $(O, L)$ there is $ε_{0} > 0$ such that $x + ε η$ is an admissible curve for $∣ ε ∣ < ε_{0}$ , the map $ε \mapsto S (x + ε η)$ is differentiable there with $δ S (x) [η] = \int_{a}^{b} [L_{x} \cdot η + L_{v} \cdot η^{'}] d t,$ the integrand evaluated at $(t, x (t), x^{'} (t))$ , and $E_{L} (x)$ is continuous on $[a, b]$ .

[F3]

Slots of the mechanical Lagrangian (The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator): $L (t, x, v) = K (v) - U (t, x)$ with $K (v) = \frac{1}{2} v \cdot M v$ ; the mass operator $M$ is linear and symmetric positive definite; $U$ does not depend on $v$ ; $K$ does not depend on $x$ .

[F4]

Calculus in the slots: the gradient of $v \mapsto \frac{1}{2} v \cdot M v$ at $v$ is $M v$ , and the derivative of $t \mapsto M x^{'} (t)$ is $M x^{''} (t)$ , by the chain rule and the componentwise derivative rules (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Linear map between vector spaces over the same field, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

Proof

technique · direct; instantiate the mathematical chart and first-variation suppliers at the mechanical Lagrangian and evaluate its two gradients

1.1F1F2given

Clause 1: the curve $q : = x$ and the variation $η$ satisfy the hypotheses of The first variation of a time-dependent action on a finite interval: admissibility, the integral formula and continuity of the Euler-Lagrange coefficient for the chart $(O, L)$ of [F1]; hence $x + ε η$ is admissible for sufficiently small $∣ ε ∣$ and the first variation exists with the formula of [F2].

1.2F3F4algebra

Gradient in the velocity slot: because $U$ does not depend on $v$ by [F3], $L_{v} (t, x, v) = \nabla_{v} K (v) = M v$ for every $(t, x, v)$ ; hence along $x$ one has $L_{v} (t, x (t), x^{'} (t)) = M x^{'} (t)$ .

1.3F3algebra

Gradient in the configuration slot: because $K$ does not depend on $x$ by [F3], $L_{x} (t, x, v) = - \nabla_{x} U (t, x)$ for every $(t, x, v)$ ; hence along $x$ the configuration gradient is $- \nabla_{x} U (t, x (t))$ .

1.4F4

Derivative of the velocity gradient: by [F4] the map $t \mapsto M x^{'} (t)$ is differentiable on $[a, b]$ with derivative $M x^{''} (t)$ .

2.1step 1.1step 1.2step 1.3algebra

Clause 1: substituting the computed gradients of steps 1.2 and 1.3 into the first-variation formula of step 1.1 gives exactly the displayed integral.

3.1F1step 1.3step 1.4algebra∎

Clause 2: by the definition of $E_{L}$ in [F1], steps 1.3 and 1.4 give $E_{L} (x) (t) = - \nabla_{x} U (t, x (t)) - M x^{''} (t)$ , which is the displayed identity.

Remarks

The identity of clause 2 is the bridge between the variational and the Newtonian descriptions: it says that the Euler-Lagrange expression of the mechanical Lagrangian is the negative of the model's acceleration residual $M x^{''} + \nabla_{x} U$ , whose vanishing is the equation of motion of the potential branch of the model.
The formula is a model statement only through the identification of the kinetic energy with the mass operator and of the potential part of the force with $- \nabla_{x} U$ ; the underlying analytic content is the cited mathematical first-variation formula. If residual forces are present, the full Cartesian equation also contains the residual term $d$ ; this item identifies only the potential-part equation.

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The stationary-action criterion and the potential-part Euler-Lagrange equation

Physical scope

The Euler-Lagrange equation of the mechanical Lagrangian of a declared potential presentation: for a C^2 curve of the model on a fixed interval, stationarity of the action, the vanishing of the Euler-Lagrange expression and the pointwise equation mass-operator acceleration equals minus the potential gradient are equivalent. The equivalence identifies the variational critical-point condition with the potential-part equation on the interval, and with the full model equation when d is identically zero; it is conditional on the adopted force branch and the declared presentation, and it asserts no solution existence, no minimisation and no empirical content.

Statement

Adopt the force model with potential presentation $(U, d)$ and the mechanical Lagrangian $L = K - U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, with masses $m_{1}, \dots, m_{N} > 0$ , open configuration domain $Q \subseteq R^{3 N}$ , time window $I_{0}$ , mass operator $M$ and $C^{2}$ potential $U : I_{0} \times Q \to R$ (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets). Let $a < b$ with $[a, b] \subseteq I_{0}$ , and let $x : [a, b] \to Q$ be a $C^{2}$ admissible curve (A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations). Then the following three conditions are equivalent:

$x$ is stationary for the action $S$ of $L$ ;
$E_{L} (x) (t) = 0$ for every $t \in [a, b]$ ;
$M x^{''} (t) = - \nabla_{x} U (t, x (t))$ for every $t \in [a, b]$ .

Condition 3 is the model's equation of motion of the potential branch (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (iv)) written along the curve for the presentation $(U, d)$ with the potential part alone: the residual forces $d$ do not appear, and the statement is therefore a statement about the potential part of the force map. No existence of curves satisfying these conditions is asserted.

Facts & Assumptions

Given: The model data, presentation and curve of the statement.

[F1]

The chart instance and first variation (The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, The first variation of a time-dependent action on a finite interval: admissibility, the integral formula and continuity of the Euler-Lagrange coefficient): $(O, L)$ with $O = I_{0} \times Q \times R^{3 N}$ is a time-dependent Lagrangian chart over $[a, b] \times Q$ , and the notions of admissible curve, variation, action, first variation and stationarity are those of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations instantiated at it.

[F2]

The stationarity theorem (Stationarity of the time-dependent action is equivalent to the vanishing of the Euler-Lagrange expression): a $C^{2}$ admissible curve $q$ of a time-dependent Lagrangian chart is stationary for the action if and only if its Euler-Lagrange expression vanishes identically on the interval.

[F3]

The Euler-Lagrange expression in model terms (The first variation of the mechanical action and the Euler-Lagrange expression in model terms): for the curve $x$ one has $E_{L} (x) (t) = - (M x^{''} (t) + \nabla_{x} U (t, x (t)))$ for every $t \in [a, b]$ .

[F4]

The model's equation of motion (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (iv), Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator): a motion of the model with force map $F$ satisfies $m_{i} x_{i}^{''} (t) = F_{i} (t, x (t), x^{'} (t))$ on its interval, equivalently $M x^{''} (t) = F (t, x (t), x^{'} (t))$ ; for the presentation $(U, d)$ of the statement the potential part of $F$ is $- \nabla_{x} U$ (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (a)).

Proof

technique · direct; combine the mathematical stationarity characterisation with the model-terms Euler-Lagrange identity and read the result through the equation of motion

1.1F1F2given

Conditions 1 and 2 are equivalent: $x$ is a $C^{2}$ admissible curve of the chart instance of [F1], so the stationarity theorem [F2] applied to that chart gives that $x$ is stationary for $S$ if and only if $E_{L} (x) (t) = 0$ for every $t \in [a, b]$ .

1.2F3algebra

Conditions 2 and 3 are equivalent: by [F3], at every $t \in [a, b]$ the value $E_{L} (x) (t)$ is the negative of $M x^{''} (t) + \nabla_{x} U (t, x (t))$ ; a vector-valued function on $[a, b]$ vanishes identically if and only if its negative does.

1.3F4given

Reading condition 3: by [F4], for the presentation $(U, d)$ the potential part of the model's force map is $- \nabla_{x} U$ , so condition 3 says that $x$ satisfies the model's equation of motion with the declared force map replaced by its potential part, i.e. the equation of the global potential model obtained from the presentation by discarding $d$ .

2.1step 1.1step 1.2step 1.3∎

Steps 1.1 and 1.2 establish the equivalence of conditions 1-3; step 1.3 records its reading as the model's equation of motion of the potential part. No existence, minimisation or empirical claim is made.

Remarks

Residual forces are excluded here. Condition 3 uses $- \nabla_{x} U$ only. When $d \equiv̸ 0$ the model's full equation of motion is $M x^{''} = F = - \nabla_{x} U + d$ , and the additional term $d$ appears on the right of condition 3; the variational reading of the full equation requires generalized forces and is treated separately on this page.
The equation is an identity of curves, not a solution statement. The equivalence is between three conditions on one given $C^{2}$ curve. Whether such curves exist through given data, and how they relate to the maximal motions of the force model, is the content of the following theorem.

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Stationary curves of the mechanical action are exactly the restrictions of the model's motions (global potential branch)

Physical scope

The stationary-action principle for the global potential branch of the force model: every motion of the model restricts to a stationary curve of the mechanical action on each closed subinterval, and conversely every C^2 stationary curve on a closed interval is the restriction of the unique maximal motion of the model through its initial datum, so that the stationary curves are exactly the restrictions of motions. The equivalence is conditional on the adopted force branch, on the declared potential presentation with no residual forces, and on the cited existence, uniqueness and continuation theorems; it asserts no global existence, no minimisation property and no empirical statement.

Statement

Adopt the force model with potential presentation $(U, d)$ and mechanical Lagrangian $L = K - U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, with masses $m_{1}, \dots, m_{N} > 0$ , open configuration domain $Q \subseteq R^{3 N}$ , time window $I_{0}$ , mass operator $M$ and $C^{2}$ potential $U : I_{0} \times Q \to R$ (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets). Assume the presentation has no residual forces, $d \equiv 0$ , so that the force map itself is $F_{i} (t, x, v) = - \nabla_{x_{i}} U (t, x)$ . Then:

motions are stationary. If $x : J \to Q$ is a motion of the model on an interval $J$ with nonempty interior (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model) and $a < b$ with $[a, b] \subseteq J \cap I_{0}$ , then the restriction $x ∣_{[a, b]}$ is stationary for the action $S$ on $[a, b]$ .
stationary curves are restrictions of motions. If $a < b$ with $[a, b] \subseteq I_{0}$ and $x : [a, b] \to Q$ is a $C^{2}$ admissible curve that is stationary for $S$ , then $x$ is the restriction to $[a, b]$ of the maximal motion of the model through the initial datum $(a, x (a), x^{'} (a))$ (whose existence and uniqueness are Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder clause 4); in particular the maximal interval of that motion contains $[a, b]$ .
identification. Consequently the $C^{2}$ curves $x : [a, b] \to Q$ that are stationary for the action are exactly the restrictions to $[a, b]$ of the motions of the model whose maximal interval contains $[a, b]$ , and the model equation satisfied is $M x^{''} = - \nabla_{x} U$ on that interval.

The statement is conditional on the adopted force branch and on the declared potential presentation with $d \equiv 0$ ; it asserts no existence of stationary curves for arbitrary endpoint data, no minimisation and no empirical value.

Facts & Assumptions

Given: The model data and presentation of the statement with $d \equiv 0$ , reals $a < b$ with $[a, b] \subseteq I_{0}$ , and the curves occurring in clauses 1-3.

[F1]

The chart instance (The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations): $(O, L)$ with $O = I_{0} \times Q \times R^{3 N}$ is a time-dependent Lagrangian chart, and stationarity of admissible curves is the stationarity of that chart.

[F2]

The identification of stationarity with the model's potential-part equation on the interval (The stationary-action criterion and the potential-part Euler-Lagrange equation): for a $C^{2}$ admissible curve $x : [a, b] \to Q$ , stationarity for $S$ , the vanishing of $E_{L} (x)$ on $[a, b]$ and the equation $M x^{''} (t) = - \nabla_{x} U (t, x (t))$ on $[a, b]$ are equivalent.

[F3]

Motions and the maximal motion (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (iv), Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (a), Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder): a motion on an interval $J$ with nonempty interior is a $C^{2}$ curve $x : J \to Q$ with $J \subseteq I_{0}$ satisfying $M x^{''} (t) = F (t, x (t), x^{'} (t))$ for all $t \in J$ , i.e. with $F_{i} = - \nabla_{x_{i}} U$ the equation $M x^{''} = - \nabla_{x} U$ ; and through every datum $(t_{0}, x_{0}, v_{0}) \in I_{0} \times Q \times R^{3 N}$ there is exactly one maximal motion, whose maximal interval $J_{\max}$ is an open interval containing $t_{0}$ and contained in $I_{0}$ , and every motion with that datum is a restriction of it. In particular two motions with the same datum agree on the intersection of their intervals.

[F4]

Continuation at a finite endpoint (Maximal Newtonian motions and the compact-continuation criterion: finite-time breakdown requires escape from every compact subset of the time-state domain, clauses 1-2): let $x_{\max} : J_{\max} \to Q$ be the maximal motion with a datum, $J_{\max} = (a_{0}, b_{0})$ , and $z_{\max} = (x_{\max}, x_{\max}^{}) : J_{\max} \to Q \times R^{3 N}$ its state curve, with time-state domain $D : = I_{0} \times Q \times R^{3 N}$ . If $b_{0} < + \infty$ , then for every compact $K \subseteq D$ and every $η > 0$ with $(b_{0} - η, b_{0}) \cap J_{\max} \neq \emptyset$ there is $t \in (b_{0} - η, b_{0}) \cap J_{\max}$ with $(t, z_{\max} (t)) \notin K$ ; equivalently, if a compact $K \subseteq D$ and $δ > 0$ satisfy $(t, z_{\max} (t)) \in K$ for all $t \in (b_{0} - δ, b_{0}) \cap J_{\max}$ , then $x_{\max}$ extends to a motion on a strictly larger interval than $J_{\max}$ , so $J_{\max}$ cannot have right endpoint $b_{0}$ .

[F5]

Compactness of a $C^{2}$ graph in the state domain (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset, Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, $C^{k}$ Euclidean maps and diffeomorphisms, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length): if $x : [c, d] \to Q$ is $C^{2}$ then the graph $K : = {(t, x (t), x^{'} (t)) : t \in [c, d]}$ is the image of the compact interval $[c, d]$ under the continuous map $t \mapsto (t, x (t), x^{'} (t))$ , hence compact, and it is contained in the open state domain $D = I_{0} \times Q \times R^{3 N}$ because $[c, d] \subseteq I_{0}$ and $x$ takes values in $Q$ .

Proof

technique · direct; read the equivalence of stationarity with the model equation from the preceding theorem, and compare a stationary curve with the maximal motion through its initial datum by uniqueness and the compact continuation criterion

1.1F1F2F3given

Clause 1: let $x : J \to Q$ be a motion with $[a, b] \subseteq J \cap I_{0}$ . By [F3] the curve $x$ satisfies $M x^{''} (t) = - \nabla_{x} U (t, x (t))$ on $J$ , in particular on $[a, b]$ . By [F3], $x$ is the restriction of the maximal motion through $(a, x (a), x^{'} (a))$ , whose open interval contains $J$ , hence $[a, b]$ . This maximal motion supplies a $C^{2}$ extension to an open interval around $[a, b]$ , so $x ∣_{[a, b]}$ is an admissible curve of [F1], and condition 3 of [F2] holds on $[a, b]$ ; hence by [F2] it is stationary for $S$ .

1.2F2F3given

Clause 2, setup: let $x : [a, b] \to Q$ be stationary. By [F2] the curve satisfies $M x^{''} (t) = - \nabla_{x} U (t, x (t))$ for every $t \in [a, b]$ ; the datum $(a, x (a), x^{'} (a))$ lies in $I_{0} \times Q \times R^{3 N}$ because $[a, b] \subseteq I_{0}$ and $x$ takes values in $Q$ . Let $x_{\max} : J_{\max} \to Q$ be the maximal motion with this datum and $J_{\max} = (α, β)$ its maximal interval; by [F3], $α < a < β$ . The restriction $x ∣_{[a, b]}$ is a motion on $[a, b]$ with the same datum, so by the uniqueness clause of [F3], $x_{\max}$ and $x$ agree on $J_{\max} \cap [a, b]$ .

2.1step 1.2algebra

The intersection $J_{\max} \cap [a, b]$ equals $[a, β)$ if $β \leq b$ and $[a, b]$ if $β > b$ , because $α < a$ ; hence agreement in step 1.2 says that $x_{\max} = x$ on $[a, \min (β, b))$ with the intersection half-open when $β \leq b$ and closed when $β > b$ .

3.1F4F5step 2.1algebra

Claim: $β > b$ . Suppose first $β < b$ . By step 2.1, $x_{\max} = x$ on $[a, β)$ ; by [F5] applied to $x ∣_{[a, β]}$ the set $K : = {(t, x (t), x^{'} (t)) : t \in [a, β]}$ is a compact subset of $D$ . Choose $η : = β - a > 0$ ; for every $t \in (β - η, β) \cap J_{\max} = (a, β)$ one has $(t, z_{\max} (t)) = (t, x (t), x^{'} (t)) \in K$ . The extension form of [F4] then extends $x_{\max}$ beyond $β$ , contradicting the maximality of $J_{\max}$ . Hence $β \geq b$ .

3.2F4F5step 2.1algebra

Claim: $β = b$ is impossible. If $β = b$ , then by step 2.1 $x_{\max} = x$ on $[a, b)$ and ${(t, x (t), x^{'} (t)) : t \in [a, b]}$ is a compact subset $K$ of $D$ by [F5]; taking $δ : = b - a > 0$ , every $t \in (b - δ, b) \cap J_{\max} = (a, b)$ satisfies $(t, z_{\max} (t)) \in K$ , so the extension form of [F4] extends $x_{\max}$ beyond $b$ , again contradicting maximality.

4.1step 2.1step 3.1step 3.2

By steps 3.1 and 3.2, $β > b$ ; with step 2.1 this gives $J_{\max} \cap [a, b] = [a, b]$ , so $[a, b] \subseteq J_{\max}$ and $x = x_{\max} ∣_{[a, b]}$ : the curve $x$ is the restriction of the maximal motion through its initial datum, which is clause 2.

5.1step 1.1step 4.1F2F3∎

Clause 3: clause 1 shows that every restriction of a motion with $[a, b] \subseteq J \cap I_{0}$ is stationary, and clause 2 shows that every stationary curve is such a restriction; the equation satisfied is $M x^{''} = - \nabla_{x} U$ by [F2] and [F3].

Remarks

What the converse uses. The passage from a stationary curve to a motion uses the local existence and uniqueness theorem, the compact continuation criterion and the compactness of a $C^{2}$ graph in the open state domain. None of these hypotheses is automatic for curves defined on open intervals with boundary singularities; the statement is confined to closed intervals inside the time window.
No global existence. Clause 2 produces the maximal motion through the datum; its maximal interval is guaranteed to contain $[a, b]$ but need not be all of $I_{0}$ . Finite-time breakdown of the model's dynamics is thereby compatible with the equivalence, which concerns only curves that exist on the given interval.

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Time-dependent coordinate maps transform the potential-part equation into the substituted Euler-Lagrange equation

Physical scope

Covariance of the model's equation of motion under a time-dependent coordinate map of the configuration domain: along a curve described by a C^3 map X with everywhere invertible configuration Jacobian, the Euler-Lagrange expression of the substituted Lagrangian is the transpose-Jacobian multiple of the inertial-chart expression, so the model's potential-part equation holds along the Cartesian description if and only if the substituted Euler-Lagrange equation holds along the coordinate description; when the chart condition holds, this is equivalent to stationarity of the substituted action. The statement is conditional on the adopted force branch, the potential presentation and the smooth invertibility of the coordinate map; it asserts nothing about constraints with fewer coordinates, which are treated separately, and no empirical claim.

Statement

Adopt the force model with potential presentation $(U, d)$ and mechanical Lagrangian $L = K - U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, with masses $m_{1}, \dots, m_{N} > 0$ , open configuration domain $Q \subseteq R^{3 N}$ , time window $I_{0}$ , mass operator $M$ and $C^{2}$ potential $U : I_{0} \times Q \to R$ (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets). Let $Ω \subseteq R \times R^{3 N}$ be open and nonempty and let $X : Ω \to Q$ be of class $C^{3}$ with $X_{q} (t, q)$ invertible for every $(t, q) \in Ω$ ( $C^{k}$ Euclidean maps and diffeomorphisms, Invertible Euclidean linear maps). Put

$\tilde{O} : = {(t, q, v) \in Ω \times R^{3 N} : (t, X (t, q), X_{t} (t, q) + X_{q} (t, q) v) \in I_{0} \times Q \times R^{3 N}},$

$\tilde{L} (t, q, v) : = L (t, X (t, q), X_{t} (t, q) + X_{q} (t, q) v) .$

Let $a < b$ and let $q : [a, b] \to R^{3 N}$ be a $C^{2}$ curve with $(t, q (t), q^{'} (t)) \in \tilde{O}$ for all $t \in [a, b]$ ; put $x (t) : = X (t, q (t))$ . Then:

$\tilde{O}$ is open, $\tilde{L}$ is of class $C^{2}$ on it, and

$E_{\tilde{L}} (q) (t) = X_{q} (t, q (t))^{T} E_{L} (x) (t) (t \in [a, b]);$

consequently $E_{\tilde{L}} (q) = 0$ on $[a, b]$ if and only if $E_{L} (x) = 0$ on $[a, b]$ , and (by The stationary-action criterion and the potential-part Euler-Lagrange equation) $q$ satisfies the substituted Euler-Lagrange equation if and only if the Cartesian description $x$ satisfies the model's potential-part equation $M x^{''} (t) = - \nabla_{x} U (t, x (t))$ on $[a, b]$ .

If in addition there is an open set $W \subseteq R^{3 N}$ with $q ([a, b]) \subseteq W$ and $[a, b] \times W \times R^{3 N} \subseteq \tilde{O}$ , then $(\tilde{O}, \tilde{L})$ is a time-dependent Lagrangian chart over $[a, b] \times W$ and: $q$ is stationary for the substituted action $\tilde{S} (q) = \int_{a}^{b} \tilde{L} (t, q, q^{'}) d t$ if and only if $x$ is stationary for $S$ .

The statement is conditional on the adopted force branch, on the declared potential presentation and on the smooth invertibility of $X$ ; the residual forces $d$ do not enter, so clause 1 concerns the potential part of the force map.

Facts & Assumptions

Given: The model data, presentation, the map $X$ and the curve $q$ of the statement, with $x = X \circ q$ .

[F1]

The covariance supplier (Coordinate covariance of the Euler-Lagrange expression under time-dependent point transformations): for $Ω \subseteq R \times R^{n}$ open, $X : Ω \to R^{n}$ of class $C^{3}$ with $X_{q}$ everywhere invertible, and $L$ of class $C^{2}$ on an open $O_{x} \subseteq R \times R^{n} \times R^{n}$ , the set $\tilde{O}$ defined as above is open, $\tilde{L}$ is $C^{2}$ on it, and for every $C^{2}$ curve $q$ with $(t, q (t), q^{'} (t)) \in \tilde{O}$ the transport identity holds along the curve; equivalently $E_{\tilde{L}} (q) = X_{q}^{T} E_{L} (x)$ in the notation of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations; and since $X_{q} (t, q (t))$ is invertible, the vanishing of one expression is equivalent to the vanishing of the other.

[F2]

The model-terms identity (The stationary-action criterion and the potential-part Euler-Lagrange equation, The first variation of the mechanical action and the Euler-Lagrange expression in model terms): for the mechanical Lagrangian of the presentation, a $C^{2}$ admissible curve $z$ of the Cartesian chart is stationary for $S$ if and only if $E_{L} (z) (t) = 0$ for all $t$ of its interval, and the latter is equivalent to $M z^{''} (t) = - \nabla_{x} U (t, z (t))$ there.

[F3]

The chart dictionary (A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations, The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, Stationarity of the time-dependent action is equivalent to the vanishing of the Euler-Lagrange expression): a pair $(O^{'}, L)$ with $O^{'} \subseteq R \times R^{3 N} \times R^{3 N}$ open and $[a, b] \times W \times R^{3 N} \subseteq O^{'}$ is a time-dependent Lagrangian chart over $[a, b] \times W$ ; stationarity means the vanishing of the first variation $δ S (q) [η]$ for all variations $η$ with $η (a) = η (b) = 0$ , and it is equivalent to the vanishing of the Euler-Lagrange expression on $[a, b]$ .

[F4]

The matrix fact (Invertible Euclidean linear maps, Linear map between vector spaces over the same field, Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes): if $A \in M_{n} (R)$ is invertible and $u \in R^{n}$ , then $A^{T} u = 0$ implies $u = 0$ , because $A^{T}$ is surjective.

Proof

technique · direct; instantiate the mathematical covariance proposition at the mechanical Lagrangian, then read the result through the model-terms identity and the chart dictionary

1.1F1given

Clause 1, structure: the Cartesian Lagrangian $L$ is $C^{2}$ on the open set $O_{x} : = I_{0} \times Q \times R^{3 N} \subseteq R \times R^{3 N} \times R^{3 N}$ (The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions), the map $X$ and the set $Ω$ satisfy the hypotheses of [F1] with $n : = 3 N$ , and the displayed $\tilde{O}$ , $\tilde{L}$ are exactly the supplier's constructions; hence $\tilde{O}$ is open, $\tilde{L}$ is $C^{2}$ , and the transport identity holds along $q$ .

2.1F1F4step 1.1algebra

Clause 1, identity and equivalence: substituting the transport identity of [F1] gives $E_{\tilde{L}} (q) (t) = X_{q} (t, q (t))^{T} E_{L} (x) (t)$ ; since $X_{q} (t, q (t))$ is invertible, [F4] gives $E_{\tilde{L}} (q) (t) = 0 \Leftrightarrow E_{L} (x) (t) = 0$ at each $t \in [a, b]$ , hence the vanishing conditions on the whole interval agree.

3.1F2step 2.1

Clause 1, model reading: by [F2] applied to the Cartesian curve $x$ , the vanishing of $E_{L} (x)$ on $[a, b]$ is equivalent to $M x^{''} (t) = - \nabla_{x} U (t, x (t))$ on $[a, b]$ ; combining with step 2.1, the substituted equation $E_{\tilde{L}} (q) = 0$ holds exactly when the Cartesian description satisfies the model's potential-part equation.

4.1F3step 2.1given∎

Clause 2: under the additional chart hypothesis, $(\tilde{O}, \tilde{L})$ is a time-dependent Lagrangian chart over $[a, b] \times W$ in the sense of [F3], and $q$ has values in $W$ and is $C^{2}$ , so $q$ is an admissible curve of that chart; by [F3], $q$ is stationary for $\tilde{S}$ if and only if $E_{\tilde{L}} (q) = 0$ on $[a, b]$ . Similarly the Cartesian curve $x$ is stationary for $S$ if and only if $E_{L} (x) = 0$ on $[a, b]$ , and step 2.1 gives the equivalence of the two vanishing conditions.

Remarks

Time-dependent maps are included. The map $X$ may depend on $t$ , so the result covers moving charts of the same type as the moving-frame construction of the preceding page: the substituted Lagrangian reads the Cartesian velocity as $X_{t} + X_{q} v$ , and the equation of motion in the moving coordinates is the Euler-Lagrange equation of $\tilde{L}$ by clause 1.
Fewer coordinates are a different construction. Here $X$ is a local diffeomorphism of the configuration domain ( $X_{q}$ invertible), so the number of coordinates equals the number of Cartesian degrees of freedom. The reduction to a parametrisation of a holonomic constraint locus, where $X$ has fewer coordinates than the ambient configuration space, is treated separately on this page.

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Total-derivative Lagrangians, the potential representative and scaling: invariant stationary curves with endpoint terms

Physical scope

Gauge properties of the mechanical action of a declared potential presentation: adding to the Lagrangian the total time derivative of a C^3 function defined on the chart time-configuration domain, with the function carrying action units, changes the action by endpoint values only and leaves the Euler-Lagrange expression, the stationary curves and the model's stationary-action equation unchanged; changing the potential representative by a sufficiently regular function of time alone is an instance and shifts the action by a constant-free endpoint term; multiplying the Lagrangian by a nonzero dimensionless constant leaves the stationary curves unchanged. The statements are conditional on the adopted force branch and presentation and assert no minimisation and no empirical content.

Statement

Adopt the force model with potential presentation $(U, d)$ and mechanical Lagrangian $L = K - U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, with masses $m_{1}, \dots, m_{N} > 0$ , open configuration domain $Q \subseteq R^{3 N}$ , time window $I_{0}$ , mass operator $M$ and $C^{2}$ potential $U : I_{0} \times Q \to R$ (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets). Let $a < b$ with $[a, b] \subseteq I_{0}$ , let $F : I_{0} \times Q \to R$ be of class $C^{3}$ with dimension $[F] = [S] = μ ℓ^{2} τ^{- 1}$ , and define

$L^{'} (t, x, v) : = L (t, x, v) + F_{t} (t, x) + F_{x} (t, x) \cdot v .$

Then:

$L^{'}$ is $C^{2}$ on $I_{0} \times Q \times R^{3 N}$ , has the same dimension as $L$ , and for every $C^{2}$ admissible curve $x : [a, b] \to Q$ one has $E_{L^{'}} (x) (t) = E_{L} (x) (t)$ for all $t \in [a, b]$ and

$S^{'} (x) = S (x) + F (b, x (b)) - F (a, x (a)),$

where $S^{'}$ is the action of $L^{'}$ ; consequently $x$ is stationary for $S^{'}$ if and only if $x$ is stationary for $S$ , and, by The stationary-action criterion and the potential-part Euler-Lagrange equation, both are equivalent to the model's potential-part equation $M x^{''} = - \nabla_{x} U$ along $x$ .

Potential representative. Let $c : I_{0} \to R$ be of class $C^{2}$ with dimension $[c] = [U] = μ ℓ^{2} τ^{- 2}$ and antiderivative $C$ (i.e. $C^{'} = c$ ). Then $\tilde{U} : = U + c$ remains of class $C^{2}$ and is the representative of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (c), with Lagrangian $\tilde{L} = K - \tilde{U}$ . Then $\tilde{L} = L + M$ where $M (t, x, v) = F_{t} + F_{x} \cdot v$ for $F (t, x) : = - C (t)$ , so clause 1 applies: the actions of $L$ and $\tilde{L}$ differ by $C (a) - C (b)$ , independently of the curve, and the stationary curves and the model's equation are unchanged.
Scaling. For every dimensionless real $λ \neq 0$ one has $E_{λ L} (x) = λ E_{L} (x)$ pointwise, so the stationary curves of the Lagrangian $λ L$ coincide with those of $L$ ; the value $λ = 0$ is excluded because the zero Lagrangian has every curve stationary.

The statement is conditional on the adopted force branch and presentation; it asserts no minimisation, no conservation law and no empirical content, and it does not remove the need to declare which representative and normalisation are used by an energy statement.

Facts & Assumptions

Given: The model data and presentation, reals $a < b$ with $[a, b] \subseteq I_{0}$ , the $C^{3}$ function $F$ of action dimension, the dimensionless number $λ \neq 0$ and a $C^{2}$ admissible curve $x : [a, b] \to Q$ .

[F1]

The mathematical gauge proposition (Adding a total time derivative and rescaling a Lagrangian: the Euler-Lagrange expression and the stationary curves): for a time-dependent Lagrangian chart $(O, L)$ over $[a, b] \times U$ and a $C^{3}$ function $F$ on an open set containing the chart time-configuration projection, the map $\tilde{L} : = L + F_{t} + F_{q} \cdot v$ is $C^{2}$ on $O$ , $E_{\tilde{L}} (q) = E_{L} (q)$ for every $C^{2}$ curve, the actions differ by $F (b, q (b)) - F (a, q (a))$ , the stationary curves coincide, and $E_{c L} = c E_{L}$ for every real $c$ with the stationary curves of $c L$ and $L$ coinciding for $c \neq 0$ while for $c = 0$ every $C^{2}$ curve is stationary for $0$ .

[F2]

The chart instance (The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations): $(O, L)$ with $O = I_{0} \times Q \times R^{3 N}$ is a time-dependent Lagrangian chart over $[a, b] \times Q$ , and $S$ , $E_{L}$ and stationarity are its notions.

[F3]

Stationarity and the model equation (The stationary-action criterion and the potential-part Euler-Lagrange equation): for a $C^{2}$ admissible curve $z$ of the chart, stationary for the action, vanishing of the Euler-Lagrange expression and the equation $M z^{''} = - \nabla_{x} U$ on $[a, b]$ are equivalent.

[F4]

The potential normalization and the chain rule (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (c), The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, $C^{k}$ Euclidean maps and diffeomorphisms, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length): replacing $U$ by $U + c (t)$ presents the same potential part; and for the displayed $F (t, x) = - C (t)$ one has $F_{t} = - c$ and $F_{x} = 0$ , so $F_{t} + F_{x} \cdot v = - c (t)$ .

Proof

technique · direct; apply the mathematical gauge proposition to the mechanical chart, and evaluate the potential-representative substitution and the scaling on the two Lagrangians

1.1F1F2given

Clause 1, structure: the chart $(O, L)$ is a time-dependent Lagrangian chart over $[a, b] \times Q$ with $O$ open by [F2], and the domain $I_{0} \times Q$ of the $C^{3}$ function $F$ is exactly the chart time-configuration projection; hence the hypotheses of [F1] are satisfied with this chart and this $F$ . Therefore $L^{'}$ is $C^{2}$ with $E_{L^{'}} (x) = E_{L} (x)$ and $S^{'} (x) = S (x) + F (b, x (b)) - F (a, x (a))$ for the curve $x$ , which is the first part of clause 1.

2.1F1F3step 1.1

Clause 1, stationarity: by step 1.1 the Euler-Lagrange expressions agree along $x$ , so $x$ is stationary for $S^{'}$ if and only if $x$ is stationary for $S$ ; by [F3] both are equivalent to $M x^{''} = - \nabla_{x} U$ along $x$ .

2.2F1F3F4step 1.1algebra

Clause 2: with $\tilde{U} = U + c$ and $\tilde{L} = K - \tilde{U}$ , one has $\tilde{L} = L - c (t)$ because the kinetic energies coincide; since $c$ is $C^{2}$ , its antiderivative $C$ is $C^{3}$ on $I_{0}$ ; by [F4] the choice $F (t, x) : = - C (t)$ gives $F_{t} + F_{x} \cdot v = - c (t)$ , so $\tilde{L} = L + M$ with $M = F_{t} + F_{x} \cdot v$ of the form required in clause 1. Applying step 1.1 to this $F$ gives that the Euler-Lagrange expressions of $L$ and $\tilde{L}$ agree and that $\tilde{S} (x) - S (x) = F (b, x (b)) - F (a, x (a)) = - C (b) + C (a)$ for every $C^{2}$ admissible curve; in particular the difference is independent of the curve, the stationary curves coincide, and the model's equation is unchanged by [F3].

3.1F1F3given∎

Clause 3: applying the scaling clause of [F1] to the chart $(O, L)$ gives $E_{λ L} = λ E_{L}$ for the given $λ \neq 0$ ; hence the vanishing sets on $[a, b]$ coincide and, by [F3], so do the stationary curves.

Remarks

The endpoint term is not a physical energy. The difference $F (b, x (b)) - F (a, x (a))$ in clause 1 is an endpoint value of the added function; it depends on the endpoints and on the representative chosen, and no energy balance is asserted from it.
Normalisation is a declared convention. Because the potential is determined only up to an independently chosen $c_{C} (t)$ on each connected component $C$ of $Q$ and the Lagrangian only up to total derivatives, every energy or action statement of the framework records the representative used; the model's equation of motion is insensitive to both choices.

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Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces

Physical scope

The dictionary from the model's Cartesian description to generalized coordinates: a C^3 configuration map that is an immersion; when the coordinate count is 3N, an everywhere-invertible Jacobian gives local coordinate charts, while a global chart onto the image additionally requires injectivity; for a regular holonomic-constraint locus the parametrization is a surjective immersion and may cover a configuration more than once. The item also defines the substituted Lagrangian and kinetic metric, generalized forces as Jacobian-transpose multiples, and the declared Lagrange equations with residual generalized forces. It defines objects and a model class only; it asserts no equivalence with the constrained equation of motion, which is the reduction theorem of this page, and no empirical claim.

Definition

Adopt the force model with potential presentation $(U, d)$ and mechanical Lagrangian $L = K - U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, with masses $m_{1}, \dots, m_{N} > 0$ , open configuration domain $Q \subseteq R^{3 N}$ , time window $I_{0}$ , mass operator $M$ and $C^{2}$ potential $U : I_{0} \times Q \to R$ (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets). Dimensions are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework; gradients and Jacobians are those of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, transposes those of The transpose $A^{T}$ of a matrix, and $C^{r}$ classes those of $C^{k}$ Euclidean maps and diffeomorphisms.

(a) Generalized-coordinate map. Let $r \geq 1$ and let $Ω \subseteq R^{r}$ be open and nonempty. A generalized-coordinate map of the model is a map $X : Ω \to Q$ of class $C^{3}$ whose derivative $D X (q)$ is injective for every $q \in Ω$ (Submersions and immersions between Euclidean open sets: an immersion); the coordinates $q = (q_{1}, \dots, q_{r})$ each carry a declared dimension $γ_{i}$ and unit fixed as part of the model declaration. When $r = 3 N$ , injectivity makes $D X (q) : R^{3 N} \to R^{3 N}$ a bijection: its kernel is ${0}$ , so rank-nullity gives image dimension $3 N$ , equal to the target dimension. The inverse is linear: if $y_{1} = D X (q) u_{1}$ and $y_{2} = D X (q) u_{2}$ , then linearity and uniqueness of preimages give $D X (q)^{- 1} (a y_{1} + b y_{2}) = a u_{1} + b u_{2}$ . Thus $D X (q)$ is invertible in the sense of Invertible Euclidean linear maps. The inverse-function theorem and its higher-regularity result (The Euclidean inverse function theorem, A local inverse of a $C^{k}$ regular map is $C^{k}$ ) make $X$ a local $C^{3}$ diffeomorphism at each point. This local property does not imply that $X$ is globally one-to-one: call $X$ a global coordinate chart onto $X (Ω)$ only when it is also injective. The other intended case is $r = 3 N - k$ for an integer $k$ with $1 \leq k \leq 3 N - 1$ , where $X$ may be a surjective immersion of a regular holonomic-constraint locus as in clause (d); injectivity is not required there. The admissible coordinate curves are the $C^{2}$ maps $q : J \to Ω$ on an interval $J \subseteq I_{0}$ with nonempty interior (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length); each such curve induces the configuration curve $x : = X \circ q : J \to Q$ .

(b) Substituted Lagrangian and kinetic metric. For a generalized-coordinate map $X$ define

$\tilde{L} : I_{0} \times Ω \times R^{r} ⟶ R, \tilde{L} (t, q, v) : = L (t, X (q), D X (q) v) = \frac{1}{2} v^{T} G (q) v - \tilde{U} (t, q),$

with the kinetic metric and the substituted potential

$G (q) : = D X (q)^{T} M D X (q), \tilde{U} (t, q) : = U (t, X (q)) .$

$G (q)$ is symmetric and positive definite, because $D X (q)$ is injective and $M$ is symmetric positive definite (Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes, Linear map between vector spaces over the same field, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ); its entry $G_{i j}$ has dimension $μ ℓ^{2} γ_{i}^{- 1} γ_{j}^{- 1}$ and the quadratic quantity $\frac{1}{2} v^{T} G (q) v$ has dimension $μ ℓ^{2} τ^{- 2}$ for every choice of coordinate dimensions. $\tilde{L}$ is of class $C^{2}$ ; the substituted action of an admissible coordinate curve $q : [a, b] \to Ω$ with $[a, b] \subseteq I_{0}$ is $\tilde{S} (q) : = \int_{a}^{b} \tilde{L} (t, q (t), q^{'} (t)) d t$ whenever the chart conventions of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions apply to $\tilde{L}$ , and the Lagrange operator of $\tilde{L}$ is

$Λ (\tilde{L}) (q) (t) : = \frac{d}{d t} (\frac{\partial \tilde{L}}{\partial v} (t, q (t), q^{'} (t))) - \frac{\partial \tilde{L}}{\partial q} (t, q (t), q^{'} (t)) \in R^{r} .$

(c) Generalized forces. Let $f : I_{0} \times Q \times R^{3 N} \to R^{3 N}$ be a force field of the model's standing regularity (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (iii)) — for instance the residual force $d$ of the presentation, or the model's full force map $F$ . Its generalized force in the coordinates $X$ is

$Q_{f} (t, q, v) : = D X (q)^{T} f (t, X (q), D X (q) v) \in R^{r};$

its $i$ -th component has the dimension of energy per unit of the coordinate $q_{i}$ , namely $μ ℓ^{2} τ^{- 2} γ_{i}^{- 1}$ : the Jacobian factor $\partial X_{a} / \partial q_{i}$ carries the dimension $ℓ γ_{i}^{- 1}$ and the force $f_{a}$ the dimension $μ ℓ τ^{- 2}$ . Thus $Q_{f} \cdot δ q$ has energy dimension; for a length-valued coordinate $Q_{f, i}$ has force dimension, and for a dimensionless angle it has torque dimension $μ ℓ^{2} τ^{- 2}$ . For the presentation $(U, d)$ write $Q^{r e s} : = Q_{d}$ for the generalized force of the residual part; when $d \equiv 0$ one has $Q^{r e s} \equiv 0$ .

(d) Constraint parametrizations. Let a regular holonomic-constraint datum $(O, k, g_{c})$ of Holonomic-constraint models of the force branch: constraint data, the constrained equation, virtual displacements and the scleronomic/rheonomic distinction be given, where $g_{c} : = G$ is the constraint map called $G$ in that definition; this separate notation distinguishes it from the kinetic matrix $G (q)$ in clause (b). Let $C \subseteq R^{3 N}$ be its time-independent configuration locus, so $Σ = {(t, x) \in O : x \in C}$ (the scleronomic case). A parametrization of the constrained configurations is a generalized-coordinate map $X : Ω \to Q$ with $r = 3 N - k$ , $X (Ω) = C \cap Q$ , and $(t, X (q)) \in O$ for all $(t, q) \in I_{0} \times Ω$ . Thus $X$ covers every constrained configuration, but a configuration may have more than one coordinate representative; a covering or other surjective immersion is allowed. Injectivity is an optional additional condition for applications that require unique coordinates. Each admissible coordinate curve induces a constrained curve family with permitted velocity; the reduction theorem of this page also identifies the virtual displacements with the image of $D X$ .

(e) Declared Lagrange equations. With the data of clauses (a)-(c), the model with presentation $(U, d)$ declares as its Lagrange equations in the coordinates $X$

$Λ (\tilde{L}) (q) (t) = Q^{r e s} (t, q (t), q^{'} (t)) (t \in J),$

posed for $C^{2}$ admissible coordinate curves $q : J \to Ω$ . These equations are part of the coordinate description: for $d \equiv 0$ they reduce to the Euler-Lagrange equation $Λ (\tilde{L}) (q) = 0$ of the substituted Lagrangian. For a chart ( $r = 3 N$ ), the transport identity of Coordinate covariance of the Euler-Lagrange expression under time-dependent point transformations clause 4, applied to the time-independent map $X$ , gives $Λ (\tilde{L}) = D X^{T} (M x^{''} - F + d)$ along $x = X \circ q$ : here $L_{v} = M x^{'}$ and $L_{x} = - \nabla_{x} U$ , while $F = - \nabla_{x} U + d$ . Hence $Λ (\tilde{L}) = D X^{T} d$ is equivalent to $M x^{''} = F$ , since $D X^{T}$ is invertible. On any interval $J$ , the transport calculation applies on compact subintervals around each interior time and extends to included endpoints by continuity of these expressions. Whether the same equations describe an ideal constrained motion for a parametrization of clause (d) is a separate statement, proved as the reduction theorem of this page; it is not part of this definition.

(f) Status. This item defines the dictionary between the Cartesian and the generalized-coordinate descriptions: the coordinate map, the substituted Lagrangian, the kinetic metric, generalized forces and the declared Lagrange equations. It adopts no force law beyond the model's presentation, asserts no solution existence and makes no empirical claim; the equivalence of the declared equations with the model's Cartesian equation (for charts) and with the ideal constrained equation (for parametrizations) is proved on this page.

Remarks

Coordinate dimensions. The kinetic metric is the coordinate expression of the fixed mass operator: with length-valued coordinates it has the dimension of mass, with dimensionless angle coordinates the angular-angular entries have dimension $μ ℓ^{2}$ , and mixed entries have the dimensions $μ ℓ^{2} γ_{i}^{- 1} γ_{j}^{- 1}$ recorded in clause (b). The physically meaningful objects are the quadratic form $v^{T} G (q) v$ and the generalized forces paired with the coordinate variations, both of which have the dimensions displayed.
The residual split enters the forces, not the Lagrangian. The substituted Lagrangian is built from $U$ alone; the residual force $d$ of the presentation appears only through $Q^{r e s}$ . A presentation with a different split of the same total force changes $Q^{r e s}$ and $L$ correspondingly, and the declared equations are unchanged, as they must be for the total force map to be the model's datum.

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Transport of the Euler-Lagrange expression under a C^3 map: E of the substituted Lagrangian is the Jacobian-transpose multiple of E of the ambient Lagrangian

Statement

Let $n, m \geq 1$ , let $Ω \subseteq R^{n}$ and $Q \subseteq R^{m}$ be open and nonempty, and let $X : Ω \to Q$ be of class $C^{3}$ ( $C^{k}$ Euclidean maps and diffeomorphisms). Let $J \subseteq R$ be an open interval and let $L : J \times Q \times R^{m} \to R$ be of class $C^{2}$ , with gradients $L_{x}$ and $L_{w}$ in the second and third slots (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ). Define

$\tilde{L} : J \times Ω \times R^{n} ⟶ R, \tilde{L} (t, q, v) : = L (t, X (q), D X (q) v) .$

Then $\tilde{L}$ is of class $C^{2}$ , and for every interval $J^{'} \subseteq J$ with nonempty interior and every $C^{2}$ curve $q : J^{'} \to Ω$ (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral) with $x : = X \circ q$ one has, writing ${\tilde{L}}_{q}$ and ${\tilde{L}}_{v}$ for the gradients of $\tilde{L}$ in the second and third slots,

${\tilde{L}}_{q} (t, q (t), q^{'} (t)) - \frac{d}{d t} ({\tilde{L}}_{v} (t, q (t), q^{'} (t))) = D X (q (t))^{T} (L_{x} (t, x (t), x^{'} (t)) - \frac{d}{d t} (L_{w} (t, x (t), x^{'} (t))))$

for every $t \in J^{'}$ . In words: the Euler-Lagrange expression of the substituted Lagrangian is the transpose-Jacobian multiple of the Euler-Lagrange expression of the ambient Lagrangian. No rank or invertibility condition on $D X$ is assumed; the identity holds for every $C^{3}$ map $X$ , square or not.

Facts & Assumptions

Given: The open sets $Ω, Q$ , the $C^{3}$ map $X : Ω \to Q$ , the $C^{2}$ function $L$ , the interval $J^{'}$ and the $C^{2}$ curve $q : J^{'} \to Ω$ with $x = X \circ q$ ; write $w (t) : = D X (q (t)) q^{'} (t)$ and $T_{i j}^{a} (q) : = \partial_{i} \partial_{j} X_{a} (q)$ , so that $X_{q q} [v]$ denotes the $m \times n$ matrix with entries $\sum_{j} T_{i j}^{a} (q) v_{j}$ , and $X_{q q} [v, v]$ the vector $(\sum_{i, j} T_{i j}^{a} v_{i} v_{j})_{a}$ .

[F1]

Differentiability of composites ( $C^{k}$ Euclidean maps are closed under componentwise algebra and composition, $C^{k}$ Euclidean maps and diffeomorphisms, $C^{k}$ maps and multi-index derivative notation in Euclidean space): $D X$ is $C^{2}$ and $(q, v) \mapsto D X (q) v$ is $C^{2}$ ; the map $Φ (t, q, v) : = (t, X (q), D X (q) v)$ is $C^{2}$ , and $\tilde{L} = L \circ Φ$ is $C^{2}$ .

[F2]

Chain rule (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative): along the $C^{2}$ curve $q$ one has $x^{'} = D X (q) q^{'} = w$ , so the curve $t \mapsto (t, x (t), w (t))$ is $C^{1}$ and, for the $C^{2}$ function $L$ , the map $t \mapsto L_{w} (t, x (t), w (t))$ is $C^{1}$ with derivative given by the chain rule; similarly $\frac{d}{d t} D X (q (t)) = X_{q q} [q^{'}] (q (t))$ in the matrix sense.

[F3]

Mixed partials (Clairaut--Schwarz theorem for continuous second partial derivatives): since $X$ is $C^{3}$ , its second partials are $C^{1}$ and $\partial_{j} \partial_{i} X_{a} = \partial_{i} \partial_{j} X_{a}$ , so the matrix $X_{q q} [q^{'}]$ is computed unambiguously and $X_{q q} [q^{'} (t)] q^{'} (t) = X_{q q} [q^{'} (t), q^{'} (t)]$ .

[F4]

Bilinear algebra of the pairing (Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes, The transpose $A^{T}$ of a matrix, Linear map between vector spaces over the same field, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ): for matrices $A$ and vectors $u$ one has $(A^{T} u) \cdot v = u \cdot (A v)$ ; sums and scalar multiples of the $C^{1}$ maps occurring below are differentiated termwise; and $\frac{d}{d t} (A (t)^{T} u (t)) = \dot{A} (t)^{T} u (t) + A (t)^{T} \dot{u} (t)$ .

Proof

technique · direct; differentiate the substituted Lagrangian in both slots by the chain rule, then cancel the parametrization-correction terms

1.1F1

Regularity: the substitution $Φ (t, q, v) = (t, X (q), D X (q) v)$ takes values in the open domain $J \times Q \times R^{m}$ because $X (Ω) \subseteq Q$ ; by [F1] the map $\tilde{L} = L \circ Φ$ is $C^{2}$ on $J \times Ω \times R^{n}$ .

1.2F1F2F4algebra

Velocity gradient: by the chain rule applied to $v \mapsto L (t, X (q), D X (q) v)$ , for every $(t, q, v)$ ${\tilde{L}}_{v} (t, q, v) = D X (q)^{T} L_{w} (t, X (q), D X (q) v) .$

1.3F1F2F4algebra

Configuration gradient: by the chain rule, for every $(t, q, v)$ ${\tilde{L}}_{q} (t, q, v) = D X (q)^{T} L_{x} (t, X (q), D X (q) v) + (X_{q q} [v])^{T} L_{w} (t, X (q), D X (q) v) .$

2.1F2F4step 1.2algebra

Derivative of the velocity gradient along the curve: from step 1.2 and the chain rule, using $\frac{d}{d t} D X (q (t)) = X_{q q} [q^{'} (t)]$ of [F2], $\frac{d}{d t} ({\tilde{L}}_{v} (t, q (t), q^{'} (t))) = X_{q q} [q^{'} (t)]^{T} L_{w} (t, x (t), w (t)) + D X (q (t))^{T} \frac{d}{d t} (L_{w} (t, x (t), w (t))) .$

3.1F2F3step 1.3step 2.1algebra

Cancellation: on the curve $v = q^{'} (t)$ , so [F3] gives $X_{q q} [q^{'} (t)]^{T} L_{w} = (X_{q q} [v])^{T} L_{w}$ at $(t, q (t), q^{'} (t))$ ; subtracting step 2.1 from step 1.3 leaves $D X (q (t))^{T} (L_{x} (t, x, x^{'}) - \frac{d}{d t} L_{w} (t, x, x^{'}))$ , which is the displayed identity.

4.1step 1.1step 3.1∎

Conclusion: the regularity of step 1.1 and the identity of step 3.1 are the two assertions of the statement, for the arbitrary $C^{3}$ map $X$ and curve $q$ .

Remarks

Use with constrained coordinates. If a map $X$ is used to describe constrained configurations, this identity relates the substituted Euler-Lagrange expression to the ambient expression after multiplication by the explicit matrix $D X (q)^{T}$ . The identity is purely mathematical: no physical model, reaction law or constraint hypothesis enters it, and no rank or dimension condition is used beyond the $C^{3}$ regularity of $X$ and the $C^{2}$ regularity of $L$ .
Comparison with the invertible case. When $D X$ is invertible the vanishing of the two expressions is equivalent. In general, the identity always sends a zero ambient expression to a zero substituted expression. The converse requires that $D X (q)^{T} u = 0$ force $u = 0$ for $u \in R^{m}$ ; this item assumes no such property.

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Ideal scleronomic holonomic constraints: the substituted Lagrange equations are equivalent to the constrained model equation

Physical scope

The elimination of ideal scleronomic holonomic constraints by parametrization: for a global C^3 surjective immersion onto the constrained configurations of a scleronomic regular holonomic datum, a C^2 coordinate curve describes an ideal constrained motion of the force model exactly when it solves the declared Lagrange equations of the substituted Lagrangian with the residual generalized force; multiple coordinate representatives of one configuration are allowed. The reaction is then uniquely determined, annihilates the virtual displacements and admits the unique continuous multiplier. The statement is conditional on the adopted force branch, the potential presentation, the constraint datum with its rank hypothesis and the ideal-reaction law; it asserts no solution existence, no uniqueness of the coordinate curve for given data and no empirical claim.

Statement

Adopt the force-branch data of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance: $N_{p} \geq 1$ labelled particles with masses $m_{1}, \dots, m_{N_{p}} > 0$ , an open time window $I_{0}$ , an open configuration domain $Q \subseteq R^{3 N_{p}}$ and a force map $F$ with the standing regularity; adopt a regular holonomic-constraint datum $(O, k, g_{c})$ of Holonomic-constraint models of the force branch: constraint data, the constrained equation, virtual displacements and the scleronomic/rheonomic distinction, where $g_{c} : = G$ is the constraint map named $G$ in that definition, and the ideal-reaction law of Adopted ideal-reaction law for holonomic constraints: the d'Alembert condition, its scope and its limits. Assume the datum is scleronomic with time-independent locus $C \subseteq R^{3 N_{p}}$ :

$Σ = {(t, x) \in O : x \in C} .$

Let $(U, d)$ be a potential presentation of $F$ in the sense of clause (a) of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, so that $F_{i} (t, x, v) = - \nabla_{x_{i}} U (t, x) + d_{i} (t, x, v)$ . Let $r : = 3 N_{p} - k$ , let $Ω \subseteq R^{r}$ be open and nonempty, and let $X : Ω \to Q$ be a parametrization of the constrained configurations in the sense of clause (d) of Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces: $X$ is of class $C^{3}$ , $D X (q)$ is injective for every $q \in Ω$ , $X (Ω) = C \cap Q$ and $(t, X (q)) \in O$ for all $(t, q) \in I_{0} \times Ω$ . Injectivity of $X$ is not assumed: the image condition requires coverage of the constrained configurations, while multiple coordinate representatives are permitted. Define the kinetic matrix, substituted potential, substituted Lagrangian and residual generalized force

$G (q) : = D X (q)^{T} M D X (q), \tilde{U} (t, q) : = U (t, X (q)), \tilde{L} (t, q, v) : = \frac{1}{2} v^{T} G (q) v - \tilde{U} (t, q),$

$Q^{r e s} (t, q, v) : = D X (q)^{T} d (t, X (q), D X (q) v) .$

Let $J \subseteq I_{0}$ be an interval with nonempty interior, let $q : J \to Ω$ be a $C^{2}$ curve and put $x : = X \circ q$ . Then the following are equivalent:

$x$ is a constrained motion of the model with an ideal reaction field: there is a continuous $R : J \to R^{3 N_{p}}$ with $M x^{''} (t) = F (t, x (t), x^{'} (t)) + R (t)$ and $R (t) \cdot δ = 0$ for every $t \in J$ and every virtual displacement $δ \in \ker D_{x} g_{c} (t, x (t))$ ;
$Λ (\tilde{L}) (q) (t) = Q^{r e s} (t, q (t), q^{'} (t))$ for every $t \in J$ , i.e. the declared Lagrange equations of Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces hold along $q$ .

If these hold, then:

the reaction is unique: $R (t) = M x^{''} (t) - F (t, x (t), x^{'} (t))$ for every $t \in J$ , and there is a unique continuous $λ : J \to R^{k}$ with $R (t) = D_{x} g_{c} (t, x (t))^{T} λ (t)$ ; if the declared constraint components have dimensions $γ_{α}$ , then $[λ_{α}] = μ ℓ^{2} τ^{- 2} / γ_{α}$ ;
the virtual displacements at $x (t)$ are exactly the image of $D X (q (t))$ , $\ker D_{x} g_{c} (t, x (t)) = D X (q (t)) (R^{r})$ ;
the statement is conditional on the adopted force branch, the presentation, the scleronomic regular constraint datum and the ideal-reaction law. No solution existence, no uniqueness of the coordinate curve for given data and no empirical content are asserted.

Facts & Assumptions

Given: The model data, the scleronomic regular constraint datum, the presentation $(U, d)$ , the parametrization $X$ , the interval $J$ and the curve $q$ with $x = X \circ q$ .

[F1]

The holonomic-constraint model (Holonomic-constraint models of the force branch: constraint data, the constrained equation, virtual displacements and the scleronomic/rheonomic distinction, Adopted ideal-reaction law for holonomic constraints: the d'Alembert condition, its scope and its limits): writing $g_{c} : = G$ for the constraint map of that supplier, a constrained curve family is a $C^{2}$ curve $x : J \to R^{3 N_{p}}$ with $(t, x (t)) \in O$ and $g_{c} (t, x (t)) = 0$ ; for a scleronomic datum the locus membership is a condition on $x$ alone and $(g_{c})_{t} = 0$ on $Σ$ for the declared presentation; a constrained motion satisfies $M x^{''} = F (t, x, x^{'}) + R$ with a continuous reaction field $R$ ; the ideal-reaction law imposes $R (t) \cdot δ = 0$ for every $δ \in \ker D_{x} g_{c} (t, x (t))$ ; the virtual displacements are that kernel.

[F2]

Virtual displacements and the parametrization (Holonomic-constraint models of the force branch: constraint data, the constrained equation, virtual displacements and the scleronomic/rheonomic distinction clause (a), For an $m \times n$ matrix $A$ , $rank (A) + \dim N (A) = n$ , Submersions and immersions between Euclidean open sets, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , $C^{k}$ Euclidean maps and diffeomorphisms): $D_{x} g_{c} (t, x)$ is surjective with $k$ rows on $Σ$ , so $\dim \ker D_{x} g_{c} (t, x) = 3 N_{p} - k = r$ ; since $g_{c} (t, X (q)) = 0$ for all $(t, q) \in I_{0} \times Ω$ , the chain rule gives $D_{x} g_{c} (t, X (q)) D X (q) = 0$ , so $D X (q) (R^{r}) \subseteq \ker D_{x} g_{c} (t, X (q))$ ; the left side has dimension $r$ because $D X (q)$ is injective; hence the two spaces are equal. No global injectivity of $X$ is used.

[F3]

The transport lemma (Transport of the Euler-Lagrange expression under a C^3 map: E of the substituted Lagrangian is the Jacobian-transpose multiple of E of the ambient Lagrangian), applied with ambient open set $Q$ , $J = I_{0}$ and $X (Ω) \subseteq Q$ : with $\tilde{L}$ the substitution of the Cartesian Lagrangian $L (t, x, w) = \frac{1}{2} w \cdot M w - U (t, x)$ through $X$ , and $E_{\tilde{L}} (q) : = {\tilde{L}}_{q} (t, q, q^{'}) - \frac{d}{d t} {\tilde{L}}_{v} (t, q, q^{'})$ , one has $E_{\tilde{L}} (q) (t) = D X (q (t))^{T} E_{L} (x) (t)$ for all $t$ of the interval, where $E_{L} (x) = L_{x} (t, x, x^{'}) - \frac{d}{d t} L_{w} (t, x, x^{'})$ ; moreover $Λ (\tilde{L}) = - E_{\tilde{L}}$ by the two definitions.

[F4]

The model-terms Euler-Lagrange expression (The first variation of the mechanical action and the Euler-Lagrange expression in model terms): for the mechanical Lagrangian of the presentation, $E_{L} (x) (t) = - (M x^{''} (t) + \nabla_{x} U (t, x (t)))$ for every $t$ of the interval of a $C^{2}$ curve; equivalently $E_{L} (x) = - (M x^{''} - F^{p})$ with $F^{p} : = - \nabla_{x} U$ the potential part of the force map (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (a)).

[F5]

The multiplier lemma (Reactions annihilating the virtual displacements are transpose multiples, with a positive definite multiplier matrix): for a surjective Jacobian $A = D_{x} g_{c} (t, x (t))$ , a vector $R (t)$ annihilating $\ker A$ has a unique representation $R (t) = A^{T} λ (t)$ , with $λ = (A M^{- 1} A^{T})^{- 1} A M^{- 1} R$ . The multiplier depends $C^{1}$ on $(t, x, R)$ under the stated $C^{2}$ constraint regularity; since $x$ is $C^{2}$ and the reaction defined from the model equation is continuous, the formula gives a continuous multiplier along the curve. The pairings are the Euclidean inner products of $R^{3 N_{p}}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes, The transpose $A^{T}$ of a matrix, Linear map between vector spaces over the same field).

Proof

technique · direct; identify the virtual displacements with the image of the parametrization, translate the ideal reaction into the transpose condition, and use the transport identity together with the model-terms Euler-Lagrange expression

1.1F1F2given

Structure of the curve: since $X$ is $C^{3}$ and $q$ is $C^{2}$ , the curve $x = X \circ q$ is $C^{2}$ with values in $X (Ω) = C \cap Q \subseteq Q$ ; for every $t \in J$ one has $(t, x (t)) \in O$ by the hypothesis on $X$ , hence $g_{c} (t, x (t)) = 0$ because $Σ = {(t, x) \in O : x \in C}$ and $x (t) \in C$ ; further $x^{'} (t) = D X (q (t)) q^{'} (t) \in D X (q (t)) (R^{r})$ , and $(g_{c})_{t} = 0$ on $Σ$ by scleronomy, so $(g_{c})_{t} (t, x (t)) + D_{x} g_{c} (t, x (t)) x^{'} (t) = 0$ by [F2]: the curve is a constrained curve family with permitted velocity.

2.1F1F2F5step 1.1algebra

Clause 1 implies the transpose condition: by [F1] the ideal reaction annihilates every element of $\ker D_{x} g_{c} (t, x (t))$ , which equals $D X (q (t)) (R^{r})$ by [F2]; hence for every $\hat{δ} \in R^{r}$ , $0 = R (t) \cdot D X (q (t)) \hat{δ} = (D X (q (t))^{T} R (t)) \cdot \hat{δ}$ by [F5]; as $\hat{δ}$ is arbitrary, $D X (q (t))^{T} R (t) = 0$ .

3.1F3F4step 2.1algebra

Clause 1 implies the Lagrange equations: from the equation of motion $M x^{''} = F + R = - \nabla_{x} U + d + R$ one has $M x^{''} + \nabla_{x} U = d + R$ , so by step 2.1 $D X^{T} (M x^{''} + \nabla_{x} U) = D X^{T} d$ . By [F4] $E_{L} (x) = - (M x^{''} + \nabla_{x} U)$ , hence $D X^{T} E_{L} (x) = - D X^{T} d = - Q^{r e s}$ ; by [F3] $E_{\tilde{L}} (q) = D X^{T} E_{L} (x)$ and $Λ (\tilde{L}) = - E_{\tilde{L}}$ , so $Λ (\tilde{L}) (q) (t) = Q^{r e s} (t, q (t), q^{'} (t))$ for every $t \in J$ .

4.1F1F2F3F4step 1.1algebra

Clause 2 implies clause 1: assume the Lagrange equations. Define $R : = M x^{''} - F (\cdot, x, x^{'})$ ; the map $R$ is continuous because $x$ is $C^{2}$ and $F$ is continuous on its domain by the standing regularity of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, and the equation $M x^{''} = F + R$ holds by construction. For the ideal condition, let $δ \in \ker D_{x} g_{c} (t, x (t))$ ; by [F2], $δ = D X (q (t)) \hat{δ}$ for some $\hat{δ}$ , so $R (t) \cdot δ = (D X^{T} R (t)) \cdot \hat{δ}$ by [F5]; and $D X^{T} R = D X^{T} (M x^{''} - F) = D X^{T} (M x^{''} + \nabla_{x} U - d) = - E_{\tilde{L}} (q) - Q^{r e s} = Λ (\tilde{L}) (q) - Q^{r e s} = 0$ by [F3], [F4] and the assumed equations. Hence $R$ is an ideal reaction and $x$ is a constrained motion in the sense of clause 1. Together with step 3.1 this proves the equivalence.

5.1F1F2F5step 2.1∎

Clauses 3 and 4: if clause 1 holds with some reaction $R$ , then $M x^{''} = F + R$ gives $R = M x^{''} - F$ uniquely, and [F5] applied at each $t$ to the surjective Jacobian $D_{x} g_{c} (t, x (t))$ and the ideal vector $R (t)$ gives the unique multiplier representation; the virtual displacement identification is [F2]. This proves clauses 3 and 4, completing the statement.

Remarks

What global coverage means. The equivalence uses $X (Ω) = C \cap Q$ , so every constrained configuration occurs as $X (q)$ . The proof is pointwise and does not require one-to-one coordinates: coverings and other multiple-representation surjective immersions are permitted. For a map whose image covers only part of the locus, the theorem applies after restricting the constraint datum to that part.
Scleronomy is used twice. Time-independence of the locus gives $(g_{c})_{t} = 0$ on $Σ$ (so parametrized curves have permitted velocity) and lets the kinetic energy be read as $\frac{1}{2} v^{T} G (q) v$ without mixed $X_{t}$ terms. The rheonomic case has additional velocity-proportional terms in $\tilde{L}$ and its own treatment through the moving-chart and constraint pages; no statement about it is made here.

RemarkRemark: AI-adaptedProof: Not applicableOpen item page →

Scope of the stationary-action formulation: no minimisation, regular mechanical Lagrangians, no general velocity-constraint equivalence and empirical status

Physical scope

Scope audit of the stationary-action formulation of the force model: stationarity is not minimisation; the mechanical Lagrangians of this framework are regular and the chart Legendre map is hyperregular, so the second-order equations determine accelerations, while degenerate Lagrangians are excluded; unrestricted-action and d'Alembert dynamics are not generally equivalent for velocity constraints, as the cited counterexample shows for its specified model and data; and the stationary-action equivalence is a conditional theorem of the model, not empirical evidence. The remark asserts no new physical claim.

Remarks

Stationarity is not minimisation. The defining condition of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions is the vanishing of the first variation for all fixed-endpoint variations, and The stationary-action criterion and the potential-part Euler-Lagrange equation identifies it with the potential-part equation, and with the full model equation when $d \equiv 0$ . It is not claimed that a stationary curve minimises the action, and no second-variation or convexity hypothesis is imposed on this page: Scope of stationarity: stationary is not minimum, and no second-variation theory is developed here records the explicit quadratic model in which stationary curves are not local minima, and the verification page of this pair checks the analogous phenomenon for an oscillator of the model class.
The mechanical branch is regular. For the mechanical Lagrangians of this page the velocity Hessian in a chart is the kinetic matrix $G (q) = D X (q)^{T} M D X (q)$ (or $M$ in Cartesian data), which is symmetric positive definite because $D X (q)$ is injective and $M$ is; A mechanical chart Lagrangian with positive definite kinetic matrix is hyperregular, with the explicit kinetic-plus-potential Hamiltonian proves that such a chart Lagrangian is hyperregular with the explicit Legendre inverse, so no failure of invertibility of the Legendre map The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity occurs on this branch and the Euler-Lagrange equations determine the coordinate accelerations. A singular velocity Hessian — for which the Legendre map has no $C^{1}$ local inverse and the second-order equations need not determine accelerations — is outside the regular mechanical branch described here; the admissible coordinate maps of the generalized-coordinate definition of this page exclude it by injectivity of $D X$ and positive definiteness of $M$ .
There is no general action equivalence for velocity constraints. The reduction theorem of this page applies to ideal holonomic constraints. For a genuine velocity constraint, the unrestricted stationary-action or augmented-action prescription can differ from the d'Alembert equation of the preceding page: The velocity constraint $z^{'} = x y^{'}$ : virtual displacements, the d'Alembert law and the vakonomic law defines both prescriptions for the normalized relation $z_{s} = x y_{s}$ (corresponding physically to ${\dot{r}}_{z} = (r_{x} / L_{0}) {\dot{r}}_{y}$ with a declared length scale) and The d'Alembert and vakonomic motions of $z^{'} = x y^{'}$ differ from the same initial data proves that their evolutions differ for the specified constraint and initial data. Therefore that model branch is not obtained by "imposing a constraint inside the action", and the examples page quantifies this discrepancy. The claim is that equivalence does not hold in general; identifying the prescriptions in another model requires its own justification.
Empirical status. The equivalence of the stationary-action condition with the model's equation of motion is a conditional theorem of the adopted force branch, the declared potential presentation and the cited mathematics: it does not establish that nature extremises or stationarises an action, and it adds no empirical content beyond the force model and its potential presentation. When the presentation has residual forces, the equation in generalized coordinates retains the transformed residual term defined by Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, while the pure action reading gives only the potential-part equation; comparing predictions with observations tests the force law and the apparatus, not the variational reformulation. Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (c) supplies the further reminder that the potential and the Lagrangian are defined only up to the declared normalization, so no numerical value of the action is a model prediction without fixing that convention.
Not asserted. No global existence of stationary curves, no theory of minimisers in function spaces, no further criteria for extrema, no free- or variable-endpoint problems, and no field-theoretic or infinite-dimensional actions are part of this page; the boundary-value non-uniqueness of the oscillator, for instance, is recorded in the verification page of the mathematics track, not here.

5 · Examples, experiments and counterexamples

None yet.

Stationary action and Lagrangian dynamics

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions

Definition

Remarks

The first variation of the mechanical action and the Euler-Lagrange expression in model terms

Statement

Facts & Assumptions

Proof

Remarks

The stationary-action criterion and the potential-part Euler-Lagrange equation

Statement

Facts & Assumptions

Proof

Remarks

Stationary curves of the mechanical action are exactly the restrictions of the model's motions (global potential branch)

Statement

Facts & Assumptions

Proof

Remarks

Time-dependent coordinate maps transform the potential-part equation into the substituted Euler-Lagrange equation

Statement

Facts & Assumptions

Proof

Remarks

Total-derivative Lagrangians, the potential representative and scaling: invariant stationary curves with endpoint terms

Statement

Facts & Assumptions

Proof

Remarks

Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces

Definition

Remarks

Transport of the Euler-Lagrange expression under a C^3 map: E of the substituted Lagrangian is the Jacobian-transpose multiple of E of the ambient Lagrangian

Statement

Facts & Assumptions

Proof

Remarks

Ideal scleronomic holonomic constraints: the substituted Lagrange equations are equivalent to the constrained model equation

Statement

Facts & Assumptions

Proof

Remarks

Scope of the stationary-action formulation: no minimisation, regular mechanical Lagrangians, no general velocity-constraint equivalence and empirical status

Remarks

5 · Examples, experiments and counterexamples

Sources