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Hamilton's equations of the mechanical presentation are equivalent to the model's equation of motion, with residual forces as momentum sources and the coordinate form on every chart

Physical scope

The equivalence between the Hamiltonian evolution of a declared potential presentation of the force branch and the model's equations of motion: on the potential branch the motions, the solutions of Hamilton's equations for the explicit branch Hamiltonian and the stationary curves of the action coincide; with residual forces the model's equation is equivalent to Hamilton's equations with the residual force as a momentum source; in a declared generalized-coordinate chart the declared Lagrange equations are equivalent to the Hamilton system, and for a scleronomic regular holonomic-constraint parametrization to the ideal constrained motion. The statement is a conditional deduction from the adopted force branch, the presentation and the cited mathematics; it makes no existence, uniqueness or 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 the data and conventions of The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta: masses $m_{1}, \dots, m_{N} > 0$ , open configuration domain $Q \subseteq R^{3 N}$ , time window $I_{0}$ , mass operator $M$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator), $C^{2}$ potential $U : I_{0} \times Q \to R$ and residual forces $d$ of the standing regularity (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance). Let $H$ be the Cartesian Hamiltonian of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, let $X : Ω \to Q$ be a generalized-coordinate map with substituted Lagrangian $\tilde{L}$ , kinetic metric $G (q) = D X (q)^{T} M D X (q)$ , substituted potential $\tilde{U} (t, q) = U (t, X (q))$ and residual generalized force $Q^{r e s} (t, q, v) = D X (q)^{T} d (t, X (q), D X (q) v)$ , and let $\tilde{H}$ be the Hamiltonian in the coordinates $X$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions). Let $J \subseteq I_{0}$ be a nonempty open interval and let $x : J \to Q$ be a $C^{2}$ curve, with $p (t) : = M x^{'} (t)$ (The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta). Then:

(potential branch) Assume $d \equiv 0$ on the state domain. The following are equivalent: (a) $x$ is a motion of the model on $J$ (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model); (b) $(x, p)$ satisfies $x^{'} (t) = H_{p} (t, x (t), p (t))$ and $p^{'} (t) = - H_{x} (t, x (t), p (t))$ for every $t \in J$ ; (c) for all reals $a < b$ with $[a, b] \subseteq J$ the restriction $x ∣_{[a, b]}$ is stationary for the action $S$ of $L$ on $[a, b]$ (A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations, The stationary-action criterion and the potential-part Euler-Lagrange equation).
(residual forces) For a $C^{2}$ curve $x : J \to Q$ with $p (t) = M x^{'} (t)$ the following are equivalent: (a) $M x^{''} (t) = - \nabla_{x} U (t, x (t)) + d (t, x (t), x^{'} (t))$ for every $t \in J$ ; (b) $x^{'} (t) = H_{p} (t, x (t), p (t))$ and $p^{'} (t) = - H_{x} (t, x (t), p (t)) + d (t, x (t), M^{- 1} p (t))$ for every $t \in J$ .
(coordinate form) Let $q : J \to Ω$ be a $C^{2}$ curve and put $p (t) : = G (q (t)) q^{'} (t)$ . Then the declared Lagrange equations of Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces clause (e) are equivalent to the Hamilton system of $\tilde{H}$ : for every $t \in J$ ,

$Λ (\tilde{L}) (q) (t) = Q^{r e s} (t, q (t), q^{'} (t)) ⟺ q^{'} (t) = {\tilde{H}}_{p} (t, q (t), p (t)), p^{'} (t) = - {\tilde{H}}_{q} (t, q (t), p (t)) + Q^{r e s} (t, q (t), q^{'} (t));$

in particular, when $d \equiv 0$ the vanishing of the Lagrange operator is equivalent to $q^{'} = {\tilde{H}}_{p}$ and $p^{'} = - {\tilde{H}}_{q}$ .

(constraint elimination) Assume in addition that $d \equiv 0$ and that $X$ is a parametrization of a scleronomic regular holonomic-constraint locus in the sense of Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces clause (d), with the ideal-reaction law of Adopted ideal-reaction law for holonomic constraints: the d'Alembert condition, its scope and its limits. Then the equivalent conditions of clause 3 with $Q^{r e s} \equiv 0$ hold for $q$ if and only if $x = X \circ q$ is a constrained motion of the model with an ideal reaction (Ideal scleronomic holonomic constraints: the substituted Lagrange equations are equivalent to the constrained model equation).

The statement is conditional on the adopted force branch, the declared presentation, the declared chart and, in clause 4, the constraint datum and ideal-reaction law; it asserts no existence or uniqueness of solutions, no conservation law and no empirical content.

Facts & Assumptions

Given: The model and presentation data, the curves $x : J \to Q$ and $q : J \to Ω$ with their momentum curves, the generalized-coordinate map $X$ with its substituted data and residual generalized force, and the Hamiltonians $H$ , $\tilde{H}$ on the inverse branches.

[F1]

Model and Lagrangian data (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, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator, Dimensions and SI unit conventions of the non-relativistic classical mechanics framework): the force map has the standing regularity, the presentation has $F_{i} = - \nabla_{x_{i}} U + d_{i}$ with $U$ of class $C^{2}$ , and the mechanical Lagrangian is $L = K - U$ with $M$ symmetric positive definite; a motion of the model is defined by $m_{i} {\ddot{x}}_{i} = F_{i} (t, x, \dot{x})$ (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model); dimensions are read by the cited units item.

[F2]

Model equivalence theorem (The stationary-action criterion and the potential-part Euler-Lagrange equation, Stationarity of the time-dependent action is equivalent to the vanishing of the Euler-Lagrange expression): for a $C^{2}$ curve on a closed interval the conditions of stationarity for the action of $L$ , vanishing of the Euler-Lagrange expression $E_{L} (x)$ , and the potential-part equation $M x^{''} = - \nabla_{x} U$ are equivalent.

[F3]

Newton equivalence (Stationary curves of the mechanical action are exactly the restrictions of the model's motions (global potential branch)): on the global potential branch the stationary curves of the action on a closed interval are exactly the restrictions of the model's motions; conversely stationary curves are restrictions of the maximal motion.

[F4]

Hamiltonian branch identities (The Hamiltonian on an inverse Legendre branch: differential, equivalence with the Euler-Lagrange equation, and the energy identity): on a hyperregular fibre-full chart with inverse branch $v$ and Hamiltonian $H = p \cdot v - L$ , one has $H_{p} = v$ , $H_{q} = - L_{q}$ and $H_{t} = - L_{t}$ ; a $C^{1}$ curve $(q, p)$ satisfies Hamilton's equations if and only if $q$ is $C^{2}$ with $E_{L} (q) = 0$ and $p = L_{v} (t, q, q^{'})$ ; and $H$ is conserved along such curves whenever $L_{t}$ vanishes along them.

[F5]

Mechanical hyperregularity and Hamiltonian form (Every potential presentation is hyperregular in its charts, with inverse branch $v = M^{- 1} p$ (or $G (q)^{- 1} p$ ) and the explicit kinetic-plus-potential Hamiltonian, The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta): the presentation is hyperregular on the Cartesian chart and on every generalized-coordinate chart, with $p = M v$ , $v = M^{- 1} p$ , $H (t, x, p) = \frac{1}{2} p \cdot M^{- 1} p + U (t, x)$ and $\tilde{H} (t, q, p) = \frac{1}{2} p \cdot G (q)^{- 1} p + \tilde{U} (t, q)$ ; the fibre derivative equals the momentum covector.

[F6]

Generalized-coordinate dictionary (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces): $\tilde{L} (t, q, v) = \frac{1}{2} v^{T} G (q) v - \tilde{U} (t, q)$ with $G$ symmetric positive definite, the Lagrange operator is $Λ (\tilde{L}) (q) = \frac{d}{d t} ({\tilde{L}}_{v} (t, q, q^{'})) - {\tilde{L}}_{q} (t, q, q^{'})$ , the generalized force of the residual part is $Q^{r e s} (t, q, v) = D X (q)^{T} d (t, X (q), D X (q) v)$ , and the declared Lagrange equations are $Λ (\tilde{L}) (q) = Q^{r e s} (t, q, q^{'})$ .

[F7]

Constraint elimination (Ideal scleronomic holonomic constraints: the substituted Lagrange equations are equivalent to the constrained model equation, 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): for a scleronomic regular holonomic-constraint datum and a parametrization of the constrained configurations, a curve $x = X \circ q$ is a constrained motion with ideal reaction if and only if $Λ (\tilde{L}) (q) = Q^{r e s} (t, q, q^{'})$ along $q$ .

[F8]

Calculus of the computations (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 $g (c) \neq 0$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , 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, Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes, The transpose $A^{T}$ of a matrix, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , $C^{k}$ Euclidean maps and diffeomorphisms): the displayed gradients, products and curve derivatives are computed by the ordinary rules for differentiable maps, sums, products and compositions, with the Euclidean pairing and matrix products of the cited items.

Proof

technique · direct; translate the stationarity/Euler-Lagrange equivalence through the hyperregular branch identities and add the residual force, then repeat the computation on the generalized-coordinate chart and read the constraint-elimination theorem

1.1F5algebra

By [F5] the presentation is hyperregular on the Cartesian chart with $H (t, x, p) = \frac{1}{2} p \cdot M^{- 1} p + U (t, x)$ , so $H_{p} (t, x, p) = M^{- 1} p$ and $H_{x} (t, x, p) = \nabla_{x} U (t, x)$ ; by [F5] the momentum curve $p = M x^{'}$ is the fibre derivative $L_{v} (t, x, x^{'})$ along $x$ , so the branch hypothesis of [F4] is met on every state of the curve.

1.2F4F5F6F8algebra

Clause 3: by [F6] the substituted Lagrangian is the mechanical chart Lagrangian $\tilde{L} = \frac{1}{2} v^{T} G (q) v - \tilde{U} (t, q)$ with $G$ symmetric positive definite and ${\tilde{L}}_{v} (t, q, v) = G (q) v$ , and by [F5] its chart is hyperregular with $p = {\tilde{L}}_{v} (t, q, q^{'}) = G (q) q^{'}$ , $\tilde{H} (t, q, p) = \frac{1}{2} p \cdot G (q)^{- 1} p + \tilde{U} (t, q)$ and ${\tilde{H}}_{p} = G (q)^{- 1} p$ , ${\tilde{H}}_{q} = - {\tilde{L}}_{q} (t, q, G (q)^{- 1} p)$ by [F4]; explicitly its $i$ -th component is $\frac{1}{2} p^{T} (\partial_{q_{i}} G (q)^{- 1}) p + \partial_{q_{i}} \tilde{U} (t, q)$ , retaining the kinetic contribution when $G$ varies, with derivatives computed using [F8]; the Lagrange operator is $Λ (\tilde{L}) (q) = \frac{d}{d t} {\tilde{L}}_{v} (t, q, q^{'}) - {\tilde{L}}_{q} (t, q, q^{'}) = - E_{\tilde{L}} (q)$ , so by [F4] applied to $\tilde{L}$ the equation $Λ (\tilde{L}) (q) = 0$ is equivalent to $q^{'} = {\tilde{H}}_{p}$ and $p^{'} = - {\tilde{H}}_{q}$ .

2.1F2F3F4step 1.1algebra

Clause 1, (a) $\Leftrightarrow$ (b): by [F3] the motions of the model on the global potential branch are exactly, through their restrictions to closed subintervals, the stationary curves of the action, and by [F2] the motion condition of clause 1(a) is $M x^{''} = - \nabla_{x} U$ on $J$ , equivalently $E_{L} (x) (t) = 0$ for all $t \in J$ ; by [F4] the pair $(x, p)$ satisfies Hamilton's equations if and only if $x$ is $C^{2}$ with $E_{L} (x) = 0$ and $p = L_{v} (t, x, x^{'})$ , which by step 1.1 is exactly $p = M x^{'}$ ; hence clause 1(a) is equivalent to clause 1(b).

2.2F1F8step 1.1algebra

Clause 2: with $p = M x^{'}$ one has $p^{'} = M x^{''}$ , $H_{p} = M^{- 1} p = x^{'}$ and $H_{x} = \nabla_{x} U$ by step 1.1, all computed with [F8]; hence $M x^{''} = - \nabla_{x} U + d (t, x, x^{'})$ holds exactly when $x^{'} = H_{p}$ and $p^{'} = - H_{x} + d (t, x, M^{- 1} p)$ , since the first identity is equivalent to $x^{'} = M^{- 1} p$ and the second is the displayed equation with $x^{'}$ replaced by $M^{- 1} p$ .

2.3F4F6F8step 1.2algebra

Clause 3 with residual force: adding $Q^{r e s}$ to both sides of the Lagrange equation adds the same term to the momentum equation, because by [F4] the momentum is $p = {\tilde{L}}_{v} (t, q, q^{'})$ and ${\tilde{H}}_{q} = - {\tilde{L}}_{q}$ on the branch; hence $Λ (\tilde{L}) (q) = Q^{r e s} (t, q, q^{'})$ is equivalent to $p^{'} = - {\tilde{H}}_{q} + Q^{r e s} (t, q, q^{'})$ together with $q^{'} = {\tilde{H}}_{p}$ , in both directions.

3.1F2step 2.1algebra

Clause 1, (c) $\Leftrightarrow$ (b): by [F2] the restriction $x ∣_{[a, b]}$ is stationary for the action of $L$ on $[a, b]$ if and only if $E_{L} (x)$ vanishes on $[a, b]$ ; $J$ is the union of its closed subintervals, so clause 1(c) holds exactly when $E_{L} (x)$ vanishes on $J$ , which by step 2.1 is equivalent to clause 1(b).

3.2F7step 2.3

Clause 4: under the stated hypotheses [F7] identifies the constrained-motion condition with the declared Lagrange equation $Λ (\tilde{L}) (q) = Q^{r e s} (t, q, q^{'})$ , and $Q^{r e s} \equiv 0$ because $d \equiv 0$ ; by step 2.3 this is equivalent to the Hamilton system of clause 3.

4.1step 2.1step 3.1step 2.2step 1.2step 2.3step 3.2∎

Collecting: clause 1 follows from steps 2.1 and 3.1, clause 2 from step 2.2, clause 3 from steps 1.2 and 2.3, and clause 4 from step 3.2.

Remarks

Residual forces are not eliminated. Clause 2 exhibits the precise sense in which a prescribed residual force modifies Hamilton's equations: it enters the momentum equation as a source, exactly as it enters the Euler-Lagrange equation through the generalized force. When $d \equiv 0$ the two descriptions are equivalent, and the residual-free hypothesis of clause 1 is essential: with $d \equiv̸ 0$ a curve of clause 1(a) need not satisfy the unforced Hamilton system of clause 1(b).
Stationarity is not minimisation. Clause 1(c) concerns stationary curves of the action in the sense of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations; the equivalence with the model's motions is the content of Stationary curves of the mechanical action are exactly the restrictions of the model's motions (global potential branch) and carries no minimisation claim. The Hamiltonian reformulation inherits this: the Hamilton system is another way of writing the model's equation, not a new variational or empirical statement.
Constraints by elimination. Clause 4 covers the constraint branch of the framework by eliminating the constraint, not by adjoining multipliers to a phase-space system; the ideal-reaction law is an adopted assumption and the eliminated chart is the substituted Lagrangian of a parametrization. The constrained Hamiltonian systems of Dirac's theory are not developed or claimed on this page.

Hamilton's equations of the mechanical presentation are equivalent to the model's equation of motion, with residual forces as momentum sources and the coordinate form on every chart

Statement

Facts & Assumptions

Proof

Remarks

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