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Evolution of phase functions along the model's Hamiltonian solutions: $d f / d t = \partial_{t} f + {f, H}$ , the first-integral criterion, and the closed bracket algebra of autonomous first integrals

Physical scope

The evolution identity of the declared potential presentation: along the model's Hamiltonian solutions the total time derivative of a phase function is its explicit time derivative plus the bracket with the branch Hamiltonian; autonomous functions with vanishing bracket are constant on trajectories, and, for an autonomous Hamiltonian, the bracket of two autonomous $C^{3}$ first integrals is again a first integral and the Hamiltonian itself is a first integral. Identification of Hamiltonian solutions with force-model motions requires $d \equiv 0$ on the state domain. The statement is a conditional deduction from the adopted force branch, the declared presentation and the cited mathematics; it asserts 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 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 Hamiltonian fields 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 bracket 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 and the branch Hamiltonian $H$ , assumed of class $C^{2}$ in $(t, z)$ on $I_{0} \times W$ (The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain 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). Let $J \subseteq I_{0}$ be a nonempty open interval and let $z : J \to W$ be a $C^{1}$ solution of the Hamiltonian system $\dot{z} (t) = X_{H_{t}} (z (t))$ , equivalently, when $d \equiv 0$ on the state domain and with $z = (x, p)$ and $p = M x^{'}$ , the momentum-phase curve of a motion of the model (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 associated first-order system, motions, maximal motions and the Newtonian flow of a force model, First-order systems, initial value problems, and solutions on intervals). Then:

(evolution identity) Let $U \subseteq J \times W$ be an open neighborhood of the solution graph ${(t, z (t)) : t \in J}$ , and let $f : U \to R$ be $C^{1}$ . For $U_{t} : = {w \in W : (t, w) \in U}$ and $f_{t} : = f (t, \cdot)$ , the function $t \mapsto f (t, z (t))$ is differentiable on $J$ and

$\frac{d}{d t} f (t, z (t)) = \partial_{t} f (t, z (t)) + {f_{t}, H_{t}} (z (t)),$

where $f_{t} : = f (t, \cdot)$ , $H_{t} : = H (t, \cdot)$ and the bracket is that 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, evaluated on the open slice $U_{t}$ . 2. (first integrals) If $f$ is autonomous and ${f, H_{t}} (z (t)) = 0$ for every $t \in J$ , then $f \circ z$ is constant on $J$ ; in particular, if in addition $H$ is autonomous ( $\partial_{t} U \equiv 0$ and $d \equiv 0$ ), then $H$ itself is constant along $z$ , the Hamiltonian form of the energy conservation of The mechanical Hamiltonian is the model's energy in the inertial chart, its balance is the energy balance, and the substituted-chart Hamiltonian is the energy of time-independent coordinates. 3. (closed algebra) Let $H$ be autonomous and let $f, g$ be autonomous $C^{3}$ phase functions with ${f, H} = 0$ and ${g, H} = 0$ everywhere on their common domain. Then ${f, g}$ is again a first integral of $H$ : ${{f, g}, H} = 0$ everywhere, and ${f, g}$ is constant along every solution of the Hamiltonian system on which it is defined.

The statement is conditional on the adopted force branch, the declared presentation and the cited mathematics; it asserts no empirical content and no existence, uniqueness or completeness of solutions.

Facts & Assumptions

Given: The potential presentation with its branch Hamiltonian, its Hamiltonian field, its bracket and a $C^{1}$ solution curve $z$ of the Hamiltonian system on the interval $J$ .

[F1]

Bracket and field (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 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 bracket of a potential presentation is bilinear, skew, a derivation in each slot, satisfies the Jacobi identity, and reproduces the fundamental brackets and the Hamiltonian field, 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 pointwise coordinate bracket is defined for $C^{1}$ inputs 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, and for $C^{2}$ functions $u, v, w$ the Jacobi identity holds by The bracket of a potential presentation is bilinear, skew, a derivation in each slot, satisfies the Jacobi identity, and reproduces the fundamental brackets and the Hamiltonian field, with each nested bracket read in that $C^{1}$ coordinate extension. In clause 3, $f, g \in C^{3}$ and $H \in C^{2}$ , so all three entries satisfy this $C^{2}$ Jacobi hypothesis; moreover ${f, g} \in C^{2}$ and is admissible in clause 2. For $C^{2}$ functions $u, v$ , ${u, v} = ω (X_{u}, X_{v}) = \nabla u^{T} J \nabla v$ in canonical coordinates, ${u, v} = - {v, u}$ , and the coordinate formula ${u, v} = \sum_{a} (u_{z^{a}} v_{z^{m + a}} - u_{z^{m + a}} v_{z^{a}})$ holds; the field of the branch Hamiltonian satisfies $\dot{z} = X_{H_{t}} (z) = J \nabla_{z} H (t, z)$ along any solution.

[F2]

Model and Hamiltonian equivalence (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 Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, The mechanical Hamiltonian is the model's energy in the inertial chart, its balance is the energy balance, and the substituted-chart Hamiltonian is the energy of time-independent coordinates, 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): when $d \equiv 0$ on the state domain, the momentum-phase curves of the model's motions are exactly the solutions of the Hamiltonian system, and with $\partial_{t} U \equiv 0$ and $d \equiv 0$ the branch Hamiltonian is constant along every motion.

[F3]

Calculus ( $C^{k}$ Euclidean maps and diffeomorphisms, 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, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , $C^{k}$ Euclidean maps are closed under componentwise algebra and composition, Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , 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, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length): the chain rule gives $\frac{d}{d t} f (t, z (t)) = \partial_{t} f (t, z (t)) + \nabla_{z} f (t, z (t)) \cdot \dot{z} (t)$ ; the Euclidean product identifies the gradient pairing with the matrix product $\nabla f^{T} \dot{z}$ ; and a differentiable function on an interval whose derivative vanishes identically is constant.

Proof

technique · direct. Differentiate the phase function along the solution by the chain rule, substitute the Hamiltonian field, identify the resulting pairing with the bracket, and read off the constancy criteria

1.1F1F3algebra

For every $t$ , $(t, z (t)) \in U$ and the slice $U_{t}$ is open in $W$ , so $f_{t}$ is a $C^{1}$ phase function near $z (t)$ . By [F3] the curve $t \mapsto f (t, z (t))$ is differentiable with $\frac{d}{d t} f (t, z (t)) = \partial_{t} f (t, z (t)) + \nabla_{z} f (t, z (t)) \cdot \dot{z} (t)$ ; by [F1] the solution satisfies $\dot{z} (t) = J \nabla_{z} H (t, z (t))$ , so $\nabla_{z} f (t, z (t)) \cdot \dot{z} (t) = \nabla f_{t} (z (t))^{T} J \nabla H_{t} (z (t)) = {f_{t}, H_{t}} (z (t))$ , using the pointwise coordinate bracket, which is defined for these $C^{1}$ and $C^{2}$ slots.

2.1step 1.1

Clause 1 is step 1.1.

2.2step 1.1F1F2F3

Clause 2: if $f$ is autonomous with ${f, H_{t}} (z (t)) = 0$ for all $t \in J$ , then by step 1.1 the derivative of $t \mapsto f (z (t))$ vanishes identically on the interval $J$ , so by [F3] that function is constant; for $f : = H$ autonomous the bracket ${H, H} = 0$ by skew-symmetry of [F1], so $H$ is constant along $z$ , which is exactly the Hamiltonian restatement of the conservation clause of [F2].

3.1step 2.2F1algebra

Clause 3: let $H$ be autonomous and let $f, g$ be autonomous $C^{3}$ functions with ${f, H} = {g, H} = 0$ identically. The functions $f, g, H$ are all $C^{2}$ , so the Jacobi identity of [F1] applies even though $H$ is not assumed $C^{3}$ : ${f, {g, H}} + {g, {H, f}} + {H, {f, g}} = 0$ , with inner brackets interpreted by the pointwise $C^{1}$ extension. The first two terms vanish identically because ${g, H} = 0$ and ${H, f} = - {f, H} = 0$ by skew-symmetry; hence ${H, {f, g}} = 0$ , i.e. ${{f, g}, H} = 0$ . Since $f, g$ are $C^{3}$ , the coordinate formula gives ${f, g} \in C^{2}$ , so clause 2 applies to it and shows it is constant along every solution on which it is defined.

4.1step 1.1step 2.2step 3.1∎

Collection: clause 1 is step 1.1, clause 2 is step 2.2 and clause 3 is step 3.1; this proves the statement.

Remarks

Time-dependent Hamiltonians. Clause 1 holds for time-dependent $H$ and time-dependent phase functions; clauses 2-3 are stated for autonomous data, where the bracket ${f, H}$ is the total derivative and the closed-algebra argument applies verbatim. For a time-dependent $H$ , ${f, H_{t}} = 0$ at each $t$ along a single trajectory is not by itself an invariant statement on the phase domain, which is why clause 3 assumes an autonomous $H$ .
Regularity. Clause 3 retains $C^{3}$ functions so that ${f, g}$ is $C^{2}$ , the class used for the first-integral bracket in its proof. The Jacobi identity of The bracket of a potential presentation is bilinear, skew, a derivation in each slot, satisfies the Jacobi identity, and reproduces the fundamental brackets and the Hamiltonian field already holds for $C^{2}$ inputs with nested brackets interpreted through the pointwise $C^{1}$ extension; clauses 1-2 use only $C^{2}$ (bracket) and $C^{1}$ (function) regularity.

Evolution of phase functions along the model's Hamiltonian solutions: $d f / d t = \partial_{t} f + {f, H}$ , the first-integral criterion, and the closed bracket algebra of autonomous first integrals

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Evolution of phase functions along the model's Hamiltonian solutions: df/dt=∂tf+{f,H}df/dt=∂t​f+{f,H}, the first-integral criterion, and the closed bracket algebra of autonomous first integrals

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Evolution of phase functions along the model's Hamiltonian solutions: $d f / d t = \partial_{t} f + {f, H}$ , the first-integral criterion, and the closed bracket algebra of autonomous first integrals