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A complete integral of the model's Hamilton-Jacobi equation produces chart Hamiltonian characteristics, Jacobi constants and local canonical $(β, α)$ labels

Physical scope

The characteristics theorem of the Hamilton-Jacobi method for a declared hyperregular potential presentation of the framework's force branch: from a complete integral of the presentation's Hamilton-Jacobi equation in a fixed momentum chart, the characteristic curves of the equation solve the chart's Hamilton equations; when $d \equiv 0$ , Cartesian characteristics are motions of the model and generalized-coordinate characteristics solve the substituted Lagrange equations; the derivatives of the complete integral in the parameters are constant along them; a local converse solves the constants equation for the configuration; and the Jacobi data are local canonical coordinates of the presentation's two-form. The statement is a conditional deduction from the adopted force branch, the declared presentation and the cited mathematics; it is local in the time, configuration and parameter variables, assumes the nondegeneracy of the mixed derivative at the base point only, and asserts no global solution, no completeness 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 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 $U$ 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), the presentation being hyperregular on its charts (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). Fix one momentum chart of the presentation (Cartesian coordinates $(t, x, p)$ , or the chart $(t, q, p)$ of a generalized-coordinate map $X : Ω \to Q$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces)), with $m$ pairs of coordinates and Hamiltonian $H$ of class $C^{2}$ on $J_{0} \times V \times R^{m}$ ; let $S : J \times U \times A \to R$ be a $C^{3}$ complete integral of the model's Hamilton-Jacobi equation at $(t_{0}, q_{0}, α_{0}) \in J \times U \times A$ in the sense of The Hamilton-Jacobi equation of a declared potential presentation: the class of action-dimension functions $S$ with $S_{t} + H (t, q, S_{q}) = 0$ in one momentum chart and A complete integral of the model's Hamilton-Jacobi equation: nondegenerate mixed derivative, momentum map and Jacobi constants ( $J \subseteq J_{0}$ , $U \subseteq V$ , $A \subseteq R^{m}$ open), and put

$β_{0} : = S_{α} (t_{0}, q_{0}, α_{0}) .$

Then:

(characteristic curves) For every $α$ sufficiently close to $α_{0}$ there is an open interval $I_{α} ∋ t_{0}$ and a $C^{1}$ curve $q : I_{α} \to U$ with $q (t_{0}) = q_{0}$ solving $q^{'} (t) = H_{p} (t, q (t), S_{q} (t, q (t), α))$ , and with $p (t) : = S_{q} (t, q (t), α)$ the pair $(q, p)$ satisfies Hamilton's equations $q^{'} = H_{p}$ , $p^{'} = - H_{q}$ on $I_{α}$ . If the chart is the Cartesian chart and $d \equiv 0$ on the state domain, then $x : = q$ is a motion of the model on $I_{α}$ and $p (t) = M x^{'} (t)$ (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta); if the chart is the chart of a generalized-coordinate map and $d \equiv 0$ , then $q$ solves the declared Lagrange equations of the substituted Lagrangian and $p (t) = G (q (t)) q^{'} (t)$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, 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).
(Jacobi constants) With $(q, p)$ as in clause 1, $β (t) : = S_{α} (t, q (t), α)$ is constant on $I_{α}$ and equal to $S_{α} (t_{0}, q_{0}, α)$ .
(local converse) For $(α, β)$ sufficiently close to $(α_{0}, β_{0})$ the equation $β = S_{α} (t, q, α)$ can be solved locally for $q = Q (t, α, β)$ , and with $p : = S_{q} (t, Q (t, α, β), α)$ the curve $t \mapsto (Q, p)$ solves Hamilton's equations; in the Cartesian chart on the potential branch it is the momentum-phase curve of a motion of the model.
(canonical labels) On the domain of the local diffeomorphism $(q, p) \mapsto (β, α)$ obtained by composing $(q, p) \mapsto (q, α)$ (solving $p = S_{q}$ ) with $(q, α) \mapsto (β, α)$ , one has $ω = d β \land d α$ at every fixed $t$ , where $ω$ is the two-form of the presentation in this chart (The canonical one-form and the two-form of a potential presentation: $λ = p \cdot d x$ on the Cartesian momentum phase chart and the induced forms on each generalized-coordinate chart); equivalently, $(β, α)$ are canonical coordinates of the presentation in the sense of clause (a) of Canonical coordinates and the canonical form of the two-form for a potential presentation: the coordinate expression $ω = \sum_{a = 1}^{m} d θ^{a} \land d θ^{m + a}$ with $m$ the chart's half-dimension, and the model's own momentum charts.

Facts & Assumptions

Given: The potential presentation with the fixed momentum chart, its $C^{2}$ Hamiltonian $H$ , the open sets $J, U, A$ , the $C^{3}$ complete integral $S$ at $(t_{0}, q_{0}, α_{0})$ , and $β_{0} = S_{α} (t_{0}, q_{0}, α_{0})$ .

[F1]

Complete-integral data (The Hamilton-Jacobi equation of a declared potential presentation: the class of action-dimension functions $S$ with $S_{t} + H (t, q, S_{q}) = 0$ in one momentum chart, A complete integral of the model's Hamilton-Jacobi equation: nondegenerate mixed derivative, momentum map and Jacobi constants): $S_{t} + H (t, q, S_{q}) = 0$ on $J \times U \times A$ ; $S$ is $C^{3}$ , so all its second derivatives used below are continuous; $\det S_{q α} (t_{0}, q_{0}, α_{0}) \neq 0$ ; and $H$ is $C^{2}$ on its open domain.

[F2]

Mathematical characteristics theorem (A complete integral produces Hamilton characteristics, the Jacobi constants, a local converse and canonical $(β, α)$ labels): applied to the $C^{2}$ function $H$ and the $C^{3}$ complete integral $S$ of [F1], it yields for the chart's Hamilton system $q^{'} = H_{p}$ , $p^{'} = - H_{q}$ the four clauses: existence of the characteristic curve $q$ with $p = S_{q}$ solving Hamilton's equations; constancy of $β = S_{α}$ along it; local solvability of $β = S_{α}$ for $q$ , with the resulting curve solving Hamilton's equations; and the identity $d q \land d p = d β \land d α$ at fixed $t$ on the domain of the local diffeomorphism $(q, p) \mapsto (β, α)$ .

[F3]

Model identification (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 clauses 1 and 3, Cartesian model motions on the potential branch and declared generalized-coordinate Lagrange equations integrate their Hamiltonian fields; the forced Cartesian field is Hamiltonian exactly when the residual force has a velocity-independent potential clause 1, The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta): on the potential branch $d \equiv 0$ and in the Cartesian chart, a $C^{2}$ curve $x$ is a motion of the model exactly when $(x, p)$ with $p (t) = M x^{'} (t)$ satisfies $x^{'} = H_{p}$ , $p^{'} = - H_{x}$ ; in a generalized-coordinate chart with $d \equiv 0$ , a $C^{2}$ curve $q$ solves the declared Lagrange equations of the substituted Lagrangian exactly when $(q, P)$ with $P (t) = G (q (t)) q^{'} (t)$ satisfies $q^{'} = {\tilde{H}}_{P}$ , $P^{'} = - {\tilde{H}}_{q}$ .

[F4]

Canonical form of the presentation's two-form (The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: $Φ_{X}^{*} λ = \sum P_{a} d q^{a}$ and $Φ_{X}^{*} ω = \sum d q^{a} \land d P_{a}$ clauses 1 and 2, Canonical coordinates and the canonical form of the two-form for a potential presentation: the coordinate expression $ω = \sum_{a = 1}^{m} d θ^{a} \land d θ^{m + a}$ with $m$ the chart's half-dimension, and the model's own momentum charts clauses (a) and (b), The canonical one-form and the two-form of a potential presentation: $λ = p \cdot d x$ on the Cartesian momentum phase chart and the induced forms on each generalized-coordinate chart): in each of the presentation's own momentum charts the two-form reads $ω = \sum_{a = 1}^{m} d q^{a} \land d p_{a}$ , so the chart coordinates are canonical coordinates, and a change of coordinates $(β, α)$ in which $ω = d β \land d α$ is a canonical chart in the sense of Canonical coordinates and the canonical form of the two-form for a potential presentation: the coordinate expression $ω = \sum_{a = 1}^{m} d θ^{a} \land d θ^{m + a}$ with $m$ the chart's half-dimension, and the model's own momentum charts clause (a).

Proof

Proof technique: direct; instantiate the mathematical characteristics theorem for the chart Hamiltonian and read its conclusions through the model identification and the canonical form of the presentation's two-form.

1.1F1F2

Clause 1 for the Hamilton system: by [F1] the hypotheses of [F2] are satisfied, so for every $α$ close to $α_{0}$ there is an open interval $I_{α} ∋ t_{0}$ and a $C^{1}$ curve $q : I_{α} \to U$ with $q (t_{0}) = q_{0}$ , $q^{'} = H_{p} (t, q, S_{q} (t, q, α))$ , and with $p : = S_{q} (t, q, α)$ the pair $(q, p)$ solves $q^{'} = H_{p}$ , $p^{'} = - H_{q}$ on $I_{α}$ .

2.1step 1.1F3

Model reading of clause 1: on the potential branch $d \equiv 0$ in the Cartesian chart, the curve $x : = q$ is $C^{1}$ with $x^{'} = H_{p}$ and $p = S_{q}$ ; the first Hamilton equation reads $M^{- 1} p = x^{'}$ , so $p (t) = M x^{'} (t)$ , and clause 1 of [F3] upgrades $x$ to a $C^{2}$ motion of the model on $I_{α}$ . In a generalized-coordinate chart with $d \equiv 0$ , put $P : = p$ ; then $q^{'} = {\tilde{H}}_{P}$ and $P^{'} = - {\tilde{H}}_{q}$ , and clause 3 of [F3] shows that $q$ solves the declared Lagrange equations and $P (t) = G (q (t)) q^{'} (t)$ .

2.2F1F2step 1.1

Clause 2: by [F2], $β (t) = S_{α} (t, q (t), α)$ is constant on $I_{α}$ and equal to $S_{α} (t_{0}, q_{0}, α)$ ; the value $β_{0} = S_{α} (t_{0}, q_{0}, α_{0})$ is the labelled constant for the characteristic through the base point.

3.1F1F2step 2.1

Clause 3: by [F2], the equation $β = S_{α} (t, q, α)$ is locally solvable for $q = Q (t, α, β)$ near $(t_{0}, α_{0}, β_{0})$ , and the curve with $p = S_{q} (t, Q, α)$ solves Hamilton's equations; the model reading is clause 1 of [F3] exactly as in step 2.1, now for the curve $Q$ .

4.1F2F4step 3.1algebra

Clause 4: by [F2] the map $(q, p) \mapsto (β, α)$ is a local diffeomorphism near the base point and $d q \land d p = d β \land d α$ holds at fixed $t$ on its domain; by [F4] the chart coordinates $(q, p)$ are canonical for the presentation's two-form, so $ω = \sum_{a} d q^{a} \land d p_{a} = d q \land d p$ there; substituting gives $ω = d β \land d α$ , and [F4] identifies $(β, α)$ as canonical coordinates of the presentation in the sense of Canonical coordinates and the canonical form of the two-form for a potential presentation: the coordinate expression $ω = \sum_{a = 1}^{m} d θ^{a} \land d θ^{m + a}$ with $m$ the chart's half-dimension, and the model's own momentum charts; the local diffeomorphism used here is the one established in step 3.1.

5.1step 1.1step 2.1step 2.2step 3.1step 4.1∎

Collection: clauses 1-4 of the statement are steps 1.1, 2.1, 2.2, 3.1 and 4.1 respectively; this proves the statement.

Remarks

Locality is exactly the content of the nondegeneracy hypothesis. The characteristic curve in clause 1 exists for parameters $α$ near $α_{0}$ and on an interval $I_{α} ∋ t_{0}$ supplied by the local theory; clause 3 is local near $(t_{0}, α_{0}, β_{0})$ . No solution is extended across a conjugate time or a caustic, and no global statement about the family of characteristics is made.
The constants carry momentum dimensions. With the reading of clause (b) of A complete integral of the model's Hamilton-Jacobi equation: nondegenerate mixed derivative, momentum map and Jacobi constants, the Jacobi constants $β = S_{α}$ have the momentum-type dimensions of the chart's canonical momenta, and clause 4 says that $(β, α)$ is a canonical chart for the presentation's two-form; this is the precise content of the textbook statement that the integration constants of a complete integral provide $2 m$ constants of the motion in canonical pairs.

A complete integral of the model's Hamilton-Jacobi equation produces chart Hamiltonian characteristics, Jacobi constants and local canonical $(β, α)$ labels

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

A complete integral of the model's Hamilton-Jacobi equation produces chart Hamiltonian characteristics, Jacobi constants and local canonical (β,α)(β,α) labels

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

A complete integral of the model's Hamilton-Jacobi equation produces chart Hamiltonian characteristics, Jacobi constants and local canonical $(β, α)$ labels