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The Hamiltonian evolution of a potential presentation preserves the two-form, the bracket and the determinant on its actual time-slice domains, also when the Hamiltonian depends on time

Physical scope

Preservation of the symplectic structure by the model's Hamiltonian evolution: on the actual time-slice domains of a $C^{2}$ branch Hamiltonian, also for a time-dependent Hamiltonian, the equal-dimensional solution maps are local canonical transformations, preserve the bracket of $C^{2}$ phase functions (with the resulting $C^{1}$ compositions read through the defined pointwise bracket extension) and have unit determinant; hence the model's potential-branch flow preserves the presentation's two-form locally in time. The statement is a conditional deduction from the adopted force branch, the declared presentation and the cited mathematics; no completeness, all-time assertion or measured value is claimed.

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 momentum charts and two-form datum of 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, the canonical coordinates 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 canonical maps of Canonical maps of a potential presentation: pullback-preserving immersions and the equal-dimensional determinant and field-transport tests and the Hamiltonian field of The Hamiltonian field of a phase function for a potential presentation: the unique field $X_{f}$ with $ι_{X_{f}} ω = d f$ , and $X_{H} = (H_{p}, - H_{x})$ for the branch Hamiltonian: the Cartesian Hamiltonian $H (t, x, p) = \frac{1}{2} p \cdot M^{- 1} p + U (t, x)$ of class $C^{2}$ on $I_{0} \times W$ , where the potential $U$ and the residual forces $d$ are those of 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 and the data are those of 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 Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, 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). Let $t_{0} \in I_{0}$ and $z_{0} \in W$ . Then there are reals $h > 0$ , $ρ > 0$ and an open neighbourhood $U_{0} \subseteq W$ of $z_{0}$ such that:

(solution map) the solutions $t \mapsto Φ (t; y)$ of the Hamiltonian system $\dot{z} (t) = X_{H_{t}} (z (t))$ through $y \in U_{0}$ at time $t_{0}$ are defined for $t \in [t_{0} - h, t_{0} + h]$ , and the solution map $Φ : [t_{0} - h, t_{0} + h] \times U_{0} \to W$ is $C^{1}$ in $y$ ; for each fixed $t$ the time- $t$ map $Φ_{t} : = Φ (t; \cdot)$ is a canonical map of the presentation between equal-dimensional phase domains, and thus a local canonical transformation on $U_{0}$ :

$D Φ_{t} (y)^{T} J D Φ_{t} (y) = J, Φ_{t}^{*} ω = ω (y \in U_{0}),$

and $\det D Φ_{t} (y) = 1$ for every $y \in U_{0}$ . 2. (bracket preservation) For every $t \in [t_{0} - h, t_{0} + h]$ and all $C^{2}$ phase functions $f, g$ on $W$ , ${f \circ Φ_{t}, g \circ Φ_{t}} = {f, g} \circ Φ_{t}$ on $U_{0}$ ; the brackets on the left are the pointwise coordinate brackets of the $C^{1}$ compositions defined in The Poisson bracket of phase functions of a potential presentation: ${f, g} = ω (X_{f}, X_{g})$ and the coordinate formula in each canonical chart. 3. (potential branch) On the potential branch ( $d \equiv 0$ ) the curves $t \mapsto Φ (t; y)$ are, through 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, the momentum-phase curves of the model's motions, so the model's Hamiltonian evolution preserves the presentation's two-form, bracket and determinant on the actual time-slice domains; the assertion is local in time, and no completeness of the flow is claimed.

The statement is conditional on the adopted force branch, the declared presentation and the cited mathematics. It requires no autonomy of $H$ : the conclusion holds for each fixed pair of times on which the solutions exist.

Facts & Assumptions

Given: The presentation with its two-form and canonical charts, a $C^{2}$ branch Hamiltonian $H$ , base data $(t_{0}, z_{0})$ , and the Hamiltonian system $\dot{z} = X_{H_{t}} (z)$ .

[F1]

Forms, field and canonicity criterion (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, 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 $Φ_{X}^{*} ω = \sum d q^{a} \land d P_{a}$ , The Hamiltonian field of a phase function for a potential presentation: the unique field $X_{f}$ with $ι_{X_{f}} ω = d f$ , and $X_{H} = (H_{p}, - H_{x})$ for the branch Hamiltonian, Canonical maps of a potential presentation: pullback-preserving immersions and the equal-dimensional determinant and field-transport tests, The Poisson bracket of phase functions of a potential presentation: ${f, g} = ω (X_{f}, X_{g})$ and the coordinate formula in each canonical chart, A $C^{1}$ map between phase domains preserves the standard two-form iff $D C^{T} J D C = J$ , and a canonical diffeomorphism carries $X_{H \circ C}$ to $X_{H}$ , The Hamiltonian field of a $C^{2}$ function on a phase domain is $J \nabla H$ , and the Poisson bracket is $\sum_{i} (\partial_{q^{i}} f \partial_{p_{i}} g - \partial_{p_{i}} f \partial_{q^{i}} g)$ , A phase domain, its canonical one-form and its standard symplectic two-form): each flow slice has source and target open subsets of the same $2 n$ -dimensional phase domain; a $C^{1}$ slice $C$ is canonical exactly when $D C^{T} J_{n} D C = J_{n}$ , and then $C^{*} ω = ω$ and it is a local diffeomorphism. The field of $H_{t}$ is $X_{H_{t}} = J_{n} \nabla H (t, \cdot)$ . For $C^{2}$ functions $f, g$ , the bracket is $\nabla f^{T} J_{n} \nabla g$ ; for $C^{1}$ inputs its pointwise coordinate extension is defined by The Poisson bracket of phase functions of a potential presentation: ${f, g} = ω (X_{f}, X_{g})$ and the coordinate formula in each canonical chart.

[F2]

ODE dependence (Continuous dependence of ODE solutions on initial data and parameters, $C^{1}$ dependence of solutions on initial data, The variational equation along an ODE solution, First-order systems, initial value problems, and solutions on intervals): for the continuous, state-locally-Lipschitz right-hand side $F (t, z) : = J \nabla H (t, z)$ there are $h > 0$ and a neighbourhood $U_{0}$ of $z_{0}$ such that the solutions through $y \in U_{0}$ at $t_{0}$ exist on $[t_{0} - h, t_{0} + h]$ , the solution map $Φ$ is $C^{1}$ in $y$ , and $Y (t) : = D_{y} Φ (t; y)$ solves the variational equation $Y^{'} = J H e s s H (t, Φ (t; y)) Y$ with $Y (t_{0}) = I$ .

[F3]

Linear algebra and calculus (The Hessian of a $C^{2}$ scalar field is symmetric, The Hessian matrix and critical points of a scalar field, Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes, The transpose $A^{T}$ of a matrix, Invertible matrices and the general linear group ${GL}_{n} (F)$ , A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit, For same-sized finite square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ , For every square matrix over a commutative ring, $\det (A^{T}) = \det (A)$ , For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix, Linear map between vector spaces over the same field, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , 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, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant): the Hessian of a $C^{2}$ function is symmetric; matrix transposes, products and inverses obey the ordinary rules; the determinant is multiplicative, $\det A^{T} = \det A$ , and $A^{T} J A = J$ forces $A$ invertible because $J$ is; derivatives of matrix-valued curves are computed entrywise, and a differentiable function on an interval with vanishing derivative is constant.

Proof

technique · direct. Differentiate the canonical matrix identity along the variational equation, transfer it to the pullback of the two-form by the canonicity criterion, and compute the bracket of composed functions

1.1F2

By [F2] the right-hand side $F (t, z) = J \nabla H (t, z)$ is continuous and $C^{1}$ in the state variable, and for the given base data there are $h > 0$ and a neighbourhood $U_{0}$ on which the solution map $Φ$ is defined and $C^{1}$ in $y$ , with $Y (t) : = D_{y} Φ (t; y)$ solving $Y^{'} = J H e s s H (t, Φ (t; y)) Y$ and $Y (t_{0}) = I$ .

2.1step 1.1F3algebra

Along that solution, $\frac{d}{d t} (Y^{T} J Y) = Y^{' T} J Y + Y^{T} J Y^{'} = Y^{T} H e s s H (t, Φ) J^{T} J Y + Y^{T} J J H e s s H (t, Φ) Y$ ; since $J^{T} J = - J^{2} = I$ and $H e s s H$ is symmetric [F3], the two terms cancel, so $t \mapsto Y (t)^{T} J Y (t)$ has vanishing derivative on the interval and equals $I^{T} J I = J$ at $t_{0}$ ; by [F3] each entry is constant, so $Y (t)^{T} J Y (t) = J$ for every $t \in [t_{0} - h, t_{0} + h]$ .

3.1step 2.1F3algebra

From $Y^{T} J Y = J$ with $J$ invertible, $Y$ is invertible ( $Y v = 0$ implies $J v = Y^{T} J Y v = 0$ , hence $v = 0$ ) and taking inverses gives $Y^{- 1} J^{- 1} Y^{- T} = J^{- 1}$ , that is $Y^{- 1} J Y^{- T} = J$ in view of $J^{- 1} = - J$ ; multiplying on the left by $Y$ and on the right by $Y^{T}$ gives $Y J Y^{T} = J$ . Moreover $\det Y (t) = 1$ : the identity $Y^{T} J Y = J$ gives $(\det Y)^{2} \det J = \det J$ , so $\det Y (t) = \pm 1$ , and $t \mapsto \det Y (t)$ is continuous with $\det Y (t_{0}) = 1$ , so the determinant is $+ 1$ throughout.

4.1step 2.1step 3.1F1

Clause 1: by step 2.1 the derivative of the time- $t$ map satisfies the matrix identity at every $y \in U_{0}$ , so by the canonicity criterion of [F1] the map $Φ_{t}$ is canonical and $Φ_{t}^{*} ω = ω$ ; by step 3.1, $\det D Φ_{t} = 1$ .

4.2step 3.1F1F3algebra

Clause 2: since $f, g$ are $C^{2}$ and $Φ_{t}$ is $C^{1}$ , both $f \circ Φ_{t}$ and $g \circ Φ_{t}$ are $C^{1}$ . Their brackets are therefore defined by the pointwise $C^{1}$ coordinate extension of The Poisson bracket of phase functions of a potential presentation: ${f, g} = ω (X_{f}, X_{g})$ and the coordinate formula in each canonical chart. The chain rule [F3] gives $\nabla (f \circ Φ_{t}) = D Φ_{t}^{T} (\nabla f) \circ Φ_{t}$ and similarly for $g$ , so the coordinate formula gives ${f \circ Φ_{t}, g \circ Φ_{t}} = (\nabla f \circ Φ_{t})^{T} D Φ_{t} J D Φ_{t}^{T} (\nabla g \circ Φ_{t}) = (\nabla f \circ Φ_{t})^{T} J (\nabla g \circ Φ_{t}) = {f, g} \circ Φ_{t}$ , using $D Φ_{t} J D Φ_{t}^{T} = J$ from step 3.1 and [F1].

5.1step 4.1step 4.2

Clause 3: on the potential branch the curves $t \mapsto Φ (t; y)$ solve the Hamiltonian system, hence are, by 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 momentum-phase curves of the model's motions; the canonicity, bracket preservation and unit determinant of clauses 1-2 are local-in-time statements on the actual time-slice domains, and no completeness is asserted.

6.1step 4.1step 4.2step 5.1∎

Collection: clause 1 is step 4.1, clause 2 is step 4.2 and clause 3 is step 5.1; this proves the statement.

Remarks

Relation to the autonomous supplier. For an autonomous branch Hamiltonian, $Φ$ is the local flow of the autonomous field $X_{H}$ , and clause 1 reproduces the matrix identity of The derivative of the Hamiltonian flow satisfies $Y^{T} J Y = J$ and $\det Y = 1$ ; the time-dependent case is proved here by the same computation for the two-time solution map, using the nonautonomous variational equation $C^{1}$ dependence of solutions on initial data.
Locality. The conclusion is asserted on the neighbourhood $U_{0}$ and the compact time interval supplied by the dependence theorem; it is not a statement about the whole phase domain, and the model's own initial-value theory decides how far the solutions extend.

The Hamiltonian evolution of a potential presentation preserves the two-form, the bracket and the determinant on its actual time-slice domains, also when the Hamiltonian depends on time

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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Sources