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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.

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources