Hamiltonian vector field

In mathematics and physics, a Hamiltonian vector field is a vector field X_H on a symplectic manifold (M, \omega) associated to a smooth function H: M \to \mathbb{R}, called the Hamiltonian, and defined by the equation \iota_{X_H} \omega = -dH, or equivalently, \omega(X_H, \cdot) = -dH.^[1] This construction arises in the study of Hamiltonian mechanics and symplectic geometry, where the flow generated by X_H describes the time evolution of a physical system conserving both the symplectic structure \omega and the energy function H.^[2] In local coordinates on \mathbb{R}^{2n} with the standard symplectic form \omega = \sum_{i=1}^n dq_i \wedge dp_i, the components of X_H are given by \frac{dq_i}{dt} = \frac{\partial H}{\partial p_i} and \frac{dp_i}{dt} = -\frac{\partial H}{\partial q_i}, yielding Hamilton's canonical equations.^[3] The Hamiltonian vector field X_H is uniquely determined by the non-degeneracy of \omega, and its integral curves lie on the level sets of H, ensuring that H is constant along the flow: \mathcal{L}_{X_H} H = 0.^[1] Moreover, X_H preserves the symplectic form, satisfying \mathcal{L}_{X_H} \omega = 0, which classifies it as a symplectic vector field.^[3] The collection of all Hamiltonian vector fields forms a Lie algebra under the Lie bracket of vector fields, isomorphic to the Lie algebra of smooth functions on M equipped with the Poisson bracket \{f, g\} = X_f(g) = \omega(X_f, X_g) = -X_g(f), satisfying [X_f, X_g] = -X_{\{f,g\}}.^[2] Not every symplectic vector field is Hamiltonian; on a connected symplectic manifold, the Hamiltonian ones correspond to exact symplectic vector fields, with the obstruction lying in the first de Rham cohomology group H^1(M; \mathbb{R}).^[3] Key examples include the harmonic oscillator on \mathbb{R}^2, where H(q, p) = \frac{p^2}{2m} + \frac{k q^2}{2} generates circular orbits via X_H = \frac{p}{m} \frac{\partial}{\partial q} - k q \frac{\partial}{\partial p}, illustrating volume-preserving and energy-conserving dynamics.^[2] In broader contexts, Hamiltonian vector fields underpin integrable systems, geometric quantization, and the study of symplectomorphisms, with applications extending to celestial mechanics, quantum field theory, and differential geometry.^[1] Their divergence-free nature in canonical coordinates ensures Liouville's theorem on phase space volume preservation.^[2]

Background Concepts

Symplectic Manifolds

A symplectic manifold is a pair (M, \omega), where M is a smooth manifold and \omega is a closed nondegenerate 2-form on M.^[4] The closedness condition means that the exterior derivative satisfies d\omega = 0, ensuring that \omega defines a cohomology class in the de Rham cohomology of M. Nondegeneracy implies that for every point p \in M, the map v \mapsto \omega_p(v, \cdot) from the tangent space T_p M to its dual T_p^* M is an isomorphism.^[4] The nondegeneracy of \omega forces the dimension of M to be even, say \dim M = 2n, as the induced pairing on tangent spaces pairs distinct directions without fixed points.^[4] A fundamental local property is given by Darboux's theorem, which states that around any point in M, there exist coordinates (q^1, \dots, q^n, p_1, \dots, p_n) such that

\omega = \sum_{i=1}^n dq^i \wedge dp_i.

This canonical local expression highlights the "standard" form of symplectic structures, independent of global topology.^[4]^[5] In classical mechanics, symplectic manifolds naturally arise as phase spaces, particularly the cotangent bundle T^*Q of a configuration manifold Q, equipped with the canonical symplectic form \omega = -d\theta. Here, \theta is the tautological 1-form defined by \theta_{(q,p)}(\delta q, \delta p) = p(\delta q), where (q,p) \in T^*Q and (\delta q, \delta p) \in T_{(q,p)}(T^*Q).^[4] This structure captures the kinematics of systems with generalized coordinates q and momenta p, providing a geometric foundation for Hamiltonian dynamics.^[5] The concept originated in classical mechanics during the late 19th century, with foundational contributions from Henri Poincaré in his studies of celestial mechanics, and was formalized in symplectic geometry by pioneers including Hermann Weyl, who coined the term "symplectic" in 1939 to describe the associated linear group.^[6]^[4]

Hamiltonian Functions

A Hamiltonian function on a symplectic manifold (M, \omega) is defined as a smooth real-valued function H: M \to \mathbb{R}.^[4] Such functions serve as scalar potentials that generate dynamics on the manifold, bridging classical mechanics with symplectic geometry.^[4] In the context of mechanics, the Hamiltonian function H typically represents the total energy of a system, comprising both kinetic and potential components, within the phase space modeled by the symplectic manifold.^[4] This formulation generalizes the energy function from Lagrangian mechanics to a geometric setting, where H encodes the conserved quantities associated with the system's symmetries via Noether's theorem.^[4] Locally, on any symplectic manifold, the Darboux theorem guarantees the existence of canonical coordinates (q^1, \dots, q^n, p_1, \dots, p_n) in which H can be expressed as a function H(q, p).^[4] These coordinates reflect the natural separation into generalized positions q and momenta p, facilitating the analysis of mechanical systems in a coordinate-dependent manner.^[7] Hamiltonian functions are inherently globally defined as smooth maps on M, but the associated dynamics distinguish between locally and globally Hamiltonian cases: global Hamiltonians correspond to exact 1-forms dH, while local ones arise when the generating 1-form is merely closed, requiring the first de Rham cohomology group H^1_{\mathrm{de\ Rham}}(M) = 0 for all symplectic vector fields to be globally Hamiltonian.^[4]

Definition and Local Expression

Abstract Definition

In the context of a symplectic manifold (M, \omega), where \omega is a closed, non-degenerate 2-form, a Hamiltonian function H: M \to \mathbb{R} defines the associated Hamiltonian vector field X_H as the unique vector field satisfying \iota_{X_H} \omega = dH, with \iota denoting the interior product. The non-degeneracy of \omega ensures the uniqueness of X_H, as it induces a bundle isomorphism \flat_\omega: TM \to T^*M given by v \mapsto \iota_v \omega, which maps vector fields bijectively to exact 1-forms and thus inverts to yield a unique vector field from the exact 1-form dH. Since H is smooth, dH is a smooth 1-form, and the isomorphism \flat_\omega is smooth, it follows that X_H is a smooth vector field. There is a common sign convention variation in the literature, where some texts define X_H via \iota_{X_H} \omega = -dH, often when using the symplectic form \omega = \sum dp_i \wedge dq^i (the negative of the standard \sum dq_i \wedge dp_i), to match conventions in physics. This choice affects the direction of the flow but preserves the underlying symplectic structure and dynamics up to time reversal.^[8]

Canonical Coordinates

In symplectic geometry, local coordinates that simplify the expression of the symplectic form are known as canonical or Darboux coordinates. On a symplectic manifold (M, \omega) of dimension $2n, Darboux's theorem guarantees the existence of a coordinate chart (U, (q^1, \dots, q^n, p_1, \dots, p_n)) around any point such that the symplectic form takes the standard expression

\omega = \sum_{i=1}^n \, dq^i \wedge dp_i.

This canonical form highlights the pairing between position-like coordinates q^i and momentum-like coordinates p_i, facilitating explicit computations of geometric objects like vector fields. The Hamiltonian vector field X_H associated to a smooth function H: M \to \mathbb{R} admits a concrete local expression in these canonical coordinates. Assuming the abstract definition via the interior product \iota_{X_H} \omega = dH, the vector field takes the form

X_H = \sum_{i=1}^n \left( \frac{\partial H}{\partial p_i} \frac{\partial}{\partial q^i} - \frac{\partial H}{\partial q^i} \frac{\partial}{\partial p_i} \right).

This expression arises from the nondegeneracy of \omega, which uniquely determines X_H for each H.^[8] To derive this, substitute the assumed coordinate form of X_H = \sum_{i=1}^n (a_i \partial/\partial q^i + b_i \partial/\partial p_i) into the defining equation and compute the interior product with \omega. The interior product yields

\iota_{X_H} \omega = \sum_{i=1}^n \left( a_i \, dp_i - b_i \, dq^i \right),

since \omega pairs the basis forms as specified. Setting this equal to dH = \sum_{i=1}^n \left( \frac{\partial H}{\partial q^i} dq^i + \frac{\partial H}{\partial p_i} dp_i \right) and equating coefficients gives a_i = \partial H / \partial p_i (from dp_i) and -b_i = \partial H / \partial q^i so b_i = -\partial H / \partial q^i (from dq^i), confirming the explicit expression. This local computation makes the intrinsic definition tangible and is valid on any Darboux chart.^[8] Under canonical transformations, which are symplectomorphisms preserving \omega, the Hamiltonian vector field transforms covariantly as a vector field. Specifically, if \phi: M \to M is a symplectomorphism, then the pushforward \phi_* X_H is the Hamiltonian vector field associated to the transformed Hamiltonian H \circ \phi^{-1}. This ensures the geometric structure remains consistent across coordinate systems.^[8]

Examples

Simple Mechanical Systems

In classical mechanics, simple systems provide intuitive illustrations of Hamiltonian vector fields, where the phase space is typically the cotangent bundle of the configuration space with canonical symplectic structure. For the one-dimensional harmonic oscillator, the Hamiltonian function is given by

H(q, p) = \frac{p^2}{2m} + \frac{1}{2} k q^2,

where q is the position, p is the momentum, m is the mass, and k is the spring constant.^[9] The corresponding Hamiltonian vector field is then

X_H = \frac{p}{m} \frac{\partial}{\partial q} - k q \frac{\partial}{\partial p},

which arises from the standard expression in canonical coordinates X_H = \frac{\partial H}{\partial p} \frac{\partial}{\partial q} - \frac{\partial H}{\partial q} \frac{\partial}{\partial p}.^[9] This vector field describes oscillatory motion in phase space, with trajectories forming closed ellipses centered at the origin. For a free particle in one dimension, the Hamiltonian simplifies to the kinetic energy term alone,

H(q, p) = \frac{p^2}{2m},

since there is no potential.^[9] The associated Hamiltonian vector field is

X_H = \frac{p}{m} \frac{\partial}{\partial q},

indicating constant velocity motion parallel to the position axis in phase space, as momentum p remains fixed while position q evolves linearly with time.^[9] In systems with central forces, such as the Kepler problem describing planetary motion under inverse-square gravitation, polar coordinates in the plane are natural, with radial position r, radial momentum p_r, and conserved angular momentum L. The Hamiltonian becomes

H(r, p_r) = \frac{p_r^2}{2m} + \frac{L^2}{2m r^2} + V(r),

where V(r) = -\frac{k}{r} for the gravitational potential with k = G m_1 m_2.^[10] The Hamiltonian vector field X_H in these coordinates generates bounded elliptical orbits for negative energies, reflecting the effective potential combining centrifugal and attractive terms. In each case, the vector field X_H generates flows along trajectories that conserve the energy represented by H.^[9]^[10]

Symplectic Geometry Examples

In symplectic geometry, coadjoint orbits provide a fundamental example of Hamiltonian vector fields arising from group actions. For a Lie group G acting on its coadjoint orbit \mathcal{O}_\lambda \subset \mathfrak{g}^* equipped with the Kostant-Kirillov-Souriau symplectic form, the action is Hamiltonian with moment map \mu: \mathcal{O}_\lambda \to \mathfrak{g}^* given by the inclusion \mu(m) = m.^[4] For an element \xi \in \mathfrak{g}, the Hamiltonian function H_\xi = \langle \mu, \xi \rangle generates the infinitesimal generator X_{H_\xi} of the action, which is the fundamental vector field X_\xi tangent to the orbit.^[4] This vector field satisfies i_{X_{H_\xi}} \omega = -dH_\xi, where \omega is the symplectic form, illustrating how Lie algebra elements produce Hamiltonian flows preserving the orbit's symplectic structure. Another illustrative example occurs on cotangent bundles, which carry a canonical symplectic structure. Consider the cotangent bundle T^*N of a smooth manifold N with the standard symplectic form \omega = -d\theta, where \theta is the tautological 1-form.^[4] The Hamiltonian H: T^*N \to \mathbb{R} defined by the quadratic form H(q, p) = \frac{1}{2} g^{ij}(q) p_i p_j, corresponding to a Riemannian metric g on N, generates the geodesic flow vector field X_H.^[4] The integral curves of X_H project to geodesics on N, and X_H satisfies Hamilton's equation i_{X_H} \omega = -dH, demonstrating the symplectic nature of geodesic dynamics in this geometric setting.^[4] Hamiltonian actions of tori or circles on symplectic manifolds yield vector fields that manifest as rotations in suitable coordinates. For instance, the circle group S^1 acts on the 2-sphere S^2 with symplectic form \omega = d\theta \wedge dh (where \theta is the azimuthal angle and h the height function) by rotations e^{it} \cdot (\theta, h) = (\theta + t, h).^[11] This action is Hamiltonian with moment map \mu = h, and the generating vector field X_h = \partial/\partial \theta satisfies i_{X_h} \omega = -dh, producing rotational flows around the vertical axis that preserve \omega.^[11] More generally, for a torus T^k acting effectively on a compact symplectic manifold (M, \omega) of dimension $2n, the action admits a moment map \mu: M \to \mathbb{R}^k if k \leq n, with each component generating a Hamiltonian vector field corresponding to circle subgroup rotations.^[4] A key non-example highlights the structural constraints: on an odd-dimensional manifold, no symplectic form exists, precluding the definition of Hamiltonian vector fields in this context.^[4] Specifically, a nondegenerate closed 2-form \omega on a manifold of dimension $2n+1 cannot exist, as the associated map TM \to T^*M induced by \omega would fail to be an isomorphism due to mismatched dimensions, rendering \omega degenerate.^[12] Consequently, the equation i_X \omega = dH admits no unique solution X for arbitrary smooth H, as the nondegeneracy required for well-defined Hamiltonian vector fields is absent.^[4]

Properties of Hamiltonian Vector Fields

Hamilton's Equations

The integral curves of a Hamiltonian vector field X_H on a symplectic manifold (M, \omega) are smooth curves \gamma: I \to M, where I is an interval in \mathbb{R}, satisfying the differential equation \gamma'(t) = X_H(\gamma(t)) for all t \in I. These curves describe the trajectories of the dynamical system generated by X_H, with the Hamiltonian function H: M \to \mathbb{R} determining the direction of motion at each point via the relation \iota_{X_H} \omega = -dH.^[7] In canonical coordinates (q^i, p_i) on a cotangent bundle T^*Q, where the symplectic form takes the standard expression \omega = \sum_i dp_i \wedge dq^i, the components of X_H are given by X_H = \sum_i \left( \frac{\partial H}{\partial p_i} \frac{\partial}{\partial q^i} - \frac{\partial H}{\partial q^i} \frac{\partial}{\partial p_i} \right). To derive Hamilton's equations, consider the action of X_H on the coordinate functions: the derivative along an integral curve \gamma(t) = (q^i(t), p_i(t)) yields \frac{d}{dt} q^i(\gamma(t)) = X_H(q^i) = \frac{\partial H}{\partial p_i} and \frac{d}{dt} p_i(\gamma(t)) = X_H(p_i) = -\frac{\partial H}{\partial q^i}, since X_H(f) = \sum_i \left( \frac{\partial H}{\partial p_i} \frac{\partial f}{\partial q^i} - \frac{\partial H}{\partial q^i} \frac{\partial f}{\partial p_i} \right) for any smooth function f.^[13]^[7] Thus, the integral curves of X_H are precisely the solutions to Hamilton's equations:

\begin{aligned} \frac{dq^i}{dt} &= \frac{\partial H}{\partial p_i}, \\ \frac{dp_i}{dt} &= -\frac{\partial H}{\partial q^i}, \end{aligned}

which form a system of $2n first-order ordinary differential equations on the $2n-dimensional phase space. This equivalence establishes Hamilton's equations as the coordinate manifestation of the flow generated by X_H.^[13]

Conservation Laws

One key property of the Hamiltonian vector field X_H on a symplectic manifold (M, \omega) is that it preserves the Hamiltonian function H along its integral curves. Specifically, the Lie derivative of H with respect to X_H vanishes: \mathcal{L}_{X_H} H = 0. This implies that the time derivative of H along the flow of X_H is zero, \frac{dH}{dt} = 0, ensuring conservation of the Hamiltonian, which often represents the total energy in mechanical systems. The proof follows directly from the definition of the Hamiltonian vector field. By definition, \iota_{X_H} \omega = -dH, so the Lie derivative \mathcal{L}_{X_H} H = X_H(H) = dH(X_H) = -\omega(X_H, X_H). Since the symplectic form \omega is skew-symmetric, \omega(X_H, X_H) = 0, hence \mathcal{L}_{X_H} H = 0. More generally, any smooth function K on M that Poisson commutes with H, i.e., \{K, H\} = 0, is conserved along the flow of X_H. Indeed, the time evolution of K satisfies \frac{dK}{dt} = \{H, K\}, so \{K, H\} = 0 implies \frac{dK}{dt} = 0. Such functions K are called constants of motion or integrals of the system. This conservation arises from symmetries via Noether's theorem in the Hamiltonian setting: if a one-parameter group of canonical transformations preserves the Hamiltonian H, it generates a conserved quantity corresponding to the infinitesimal generator of the symmetry.

Symplectomorphism Generation

A Hamiltonian vector field X_H on a symplectic manifold (M, \omega) is defined such that its contraction with the symplectic form satisfies \iota_{X_H} \omega = -dH, where H is the Hamiltonian function. This ensures that X_H preserves the symplectic structure infinitesimally, as the Lie derivative of \omega along X_H vanishes: \mathcal{L}_{X_H} \omega = 0.^[1]^[14] To see this, apply Cartan's magic formula for the Lie derivative of a differential form:

\mathcal{L}_{X_H} \omega = \iota_{X_H} d\omega + d(\iota_{X_H} \omega).

Since \omega is closed, d\omega = 0, so the first term is zero. The second term simplifies to d(-dH) = -d^2 H = 0, as the exterior derivative of an exact form is zero. Thus, \mathcal{L}_{X_H} \omega = 0, confirming that X_H is a symplectic vector field.^[1]^[15]^[14] This condition positions X_H as an infinitesimal generator of symplectomorphisms, tangent to the symplectomorphism group \mathrm{Sympl}(M, \omega) at the identity. The space of all such symplectic vector fields forms a Lie algebra under the Lie bracket, with Hamiltonian vector fields comprising a distinguished subspace.^[14]^[15] A key consequence is Liouville's theorem, which states that the flow of X_H preserves the Liouville volume form \frac{\omega^n}{n!} on the $2n-dimensional manifold M. This follows because

\mathcal{L}_{X_H} \left( \frac{\omega^n}{n!} \right) = \frac{1}{n!} \left( \mathcal{L}_{X_H} \omega \right) \wedge \omega^{n-1} + \cdots + \frac{1}{n!} \omega^n \wedge \left( \mathcal{L}_{X_H} \omega \right) = 0,

since each term involves \mathcal{L}_{X_H} \omega = 0. This volume preservation is fundamental in classical mechanics for understanding phase space incompressibility.^[1]^[14]

Poisson Brackets

Definition via Vector Fields

In the context of a symplectic manifold (M, \omega), where \omega is a closed non-degenerate 2-form, the Poisson bracket of two smooth functions f, g \in C^\infty(M) is defined using the associated Hamiltonian vector fields X_f and X_g. These vector fields are uniquely determined by the relations \iota_{X_f} \omega = -df and \iota_{X_g} \omega = -dg, where \iota denotes the interior product and d is the exterior derivative.^[1] The Poisson bracket is then given by

\{f, g\} = \omega(X_f, X_g),

which establishes a bilinear operation on the space of smooth functions that encodes the symplectic structure algebraically.^[1] This definition highlights the role of Hamiltonian vector fields in translating the geometric data of \omega into an algebraic bracket operation. The Poisson bracket satisfies several fundamental properties derived directly from the symplectic form and the linearity of the interior product. It is bilinear over \mathbb{R}, meaning that for constants a, b \in \mathbb{R} and functions g, h \in C^\infty(M),

\{f, ag + bh\} = a \{f, g\} + b \{f, h\},

and similarly in the second argument.^[16] It is also skew-symmetric:

\{f, g\} = -\{g, f\},

which follows from the antisymmetry of \omega.^[1] Additionally, it obeys the Leibniz rule, acting as a derivation in each argument:

\{f, gh\} = g \{f, h\} + h \{f, g\},

reflecting the product rule for the underlying vector fields.^[16] These properties position the Poisson bracket as a key tool for analyzing the algebraic structure induced by the symplectic geometry. An important equivalence relates the Poisson bracket to the action of Hamiltonian vector fields as derivations. Specifically,

\{f, g\} = X_f(g) = -X_g(f),

where X_g(f) denotes the directional derivative of f along X_g.^[1] This identity underscores that the bracket measures how one Hamiltonian vector field differentiates another function, providing a coordinate-free interpretation of the operation. This definition arises naturally from the symplectic form via the interior product relations. Contracting \iota_{X_g} \omega = -dg with X_f yields \omega(X_g, X_f) = -dg(X_f) = -X_f(g), and since \{f, g\} = X_f(g) by the equivalence above while \omega(X_f, X_g) = - \omega(X_g, X_f), the bracket aligns directly with \omega(X_f, X_g).^[1] Symmetrically, applying \iota_{X_f} \omega = -df gives \omega(X_f, X_g) = -df(X_g) = -X_g(f), ensuring the operation is well-defined and independent of local coordinates.^[16]

Lie Algebra Structure

The Poisson bracket on the space of smooth functions C^\infty(M) on a symplectic manifold (M, \omega) defines a Lie algebra structure, with the bracket satisfying bilinearity, antisymmetry, and the Jacobi identity. Specifically, for any f, g, h \in C^\infty(M),

\{f, \{g, h\}\} + \{g, \{h, f\}\} + \{h, \{f, g\}\} = 0.

This identity holds because the Poisson bracket is derived from the symplectic form, ensuring the algebraic consistency required for a Lie bracket. The Jacobi identity for the Poisson bracket directly implies a corresponding structure on the Hamiltonian vector fields. The canonical map associating each function f to its Hamiltonian vector field X_f, defined by \iota_{X_f} \omega = -df, yields the relation [X_f, X_g] = -X_{\{f, g\}}, where [ \cdot, \cdot ] denotes the Lie bracket of vector fields. This equality follows from the definition of the Poisson bracket as \{f, g\} = X_f(g) = \omega(X_f, X_g) and the properties of the interior product and Lie derivative, confirming that the image of Hamiltonian vector fields forms a Lie subalgebra of the Lie algebra of all smooth vector fields on M. The map \phi: C^\infty(M) \to \mathfrak{X}(M), \phi(f) = X_f, is a Lie algebra anti-homomorphism from (C^\infty(M), \{ \cdot, \cdot \}) to the Lie algebra of vector fields, with kernel consisting precisely of the constant functions, as constant functions generate the zero vector field. Thus, it induces a Lie algebra isomorphism between the quotient C^\infty(M)/\mathbb{R} (with the induced bracket) and the Lie algebra \mathfrak{ham}(M, \omega) of Hamiltonian vector fields. The adjoint representation in this Lie algebra acts infinitesimally on functions via \mathrm{ad}_f(g) = \{f, g\}, mirroring the action of the Lie bracket on the vector fields through the identification.

Hamiltonian Flows

Local Flows

The local flow of a Hamiltonian vector field X_H on a symplectic manifold (M, \omega) is defined as a one-parameter family of diffeomorphisms \phi_t: U \to M, where U \subset M \times \mathbb{R} is an open set containing the zero section \{(x, 0) \mid x \in M\}, satisfying the initial value problem

\frac{d}{dt} \phi_t(x) = X_H(\phi_t(x)), \quad \phi_0(x) = x

for all x \in M.^[17] This flow describes the integral curves of X_H, which locally evolve points along the direction specified by the Hamiltonian H. The diffeomorphisms \phi_t preserve the symplectic structure locally, as the Lie derivative \mathcal{L}_{X_H} \omega = 0 ensures that each \phi_t pulls back \omega to itself.^[3] By the standard Picard-Lindelöf theorem for ordinary differential equations on manifolds, local existence and uniqueness of the flow hold whenever X_H is smooth, which it is since H is assumed smooth and \omega is nondegenerate.^[2] Specifically, for each initial point x \in M, there exists a maximal time interval I_x = (a_x, b_x) with $0 \in I_x and b_x > 0 such that the solution \phi_t(x) is defined and unique for t \in I_x, remaining within a compact subset of M where X_H is Lipschitz continuous. The flow is compact in the sense that it is defined only on this maximal interval, beyond which the solution may escape any compact set or approach the boundary of the domain if M is not complete.^[18] In the time-dependent case, where the Hamiltonian H = H(t, \cdot) varies with time, the associated vector field X_{H(t)} becomes time-dependent, generating a non-autonomous flow \phi_t satisfying

\frac{d}{dt} \phi_t(x) = X_{H(t)}(\phi_t(x)), \quad \phi_0(x) = x.

Local existence and uniqueness still follow from standard ODE theory, provided H(t, \cdot) is smooth in its spatial arguments uniformly in t over compact time intervals. The maximal interval of definition depends on both the spatial domain and the time variation of H(t), potentially shortening if the time dependence causes rapid growth in X_{H(t)}. This setup arises naturally in perturbed mechanical systems, where external time-varying forces modify the energy function.

Global Hamiltonian Vector Fields

A Hamiltonian vector field X_H on a symplectic manifold (M, \omega) generates a global flow if it is complete, meaning the flow \phi_t^H: M \to M is defined for all t \in \mathbb{R} and every point in M. Completeness ensures that integral curves extend indefinitely without singularities or escape in finite time. On compact symplectic manifolds, every smooth Hamiltonian vector field is complete. This follows from the general fact that any smooth vector field on a compact manifold generates a complete flow, as compactness bounds the trajectory and prevents finite-time blow-up. Since Hamiltonian vector fields arise from smooth Hamiltonians, and compact manifolds imply bounded Hamiltonians, the flows remain confined within the manifold for all time. Cotangent bundles T^*Q of smooth manifolds Q provide prominent examples of spaces supporting Hamiltonian vector fields with global flows. Equipped with the canonical symplectic form \omega = -d\theta, where \theta is the tautological 1-form, smooth functions on T^*Q—such as mechanical Hamiltonians—yield globally defined X_H. For typical mechanical Hamiltonians on T^*Q with compact Q, the resulting X_H often generate complete flows.^[19] The Arnold-Liouville theorem illustrates global structure in integrable systems: for a completely integrable Hamiltonian system on a $2n-dimensional symplectic manifold with n independent commuting Hamiltonians, if a regular level set is compact and connected, it is diffeomorphic to an n-torus, and global action-angle coordinates exist on a saturated neighborhood, linearizing the flows to constant speeds on the tori and ensuring completeness thereon.