Symplectic Vector Fields and Hamiltonian Flows

DefinitionSymplectic Vector Field

A vector field $X$ on a symplectic manifold $(M, \omega)$ is symplectic (or locally Hamiltonian) if its flow $\phi_t$ preserves the symplectic form:

$\mathcal{L}_X \omega = 0$

where $\mathcal{L}_X$ denotes the Lie derivative. Equivalently, using Cartan's formula:

$d(\iota_X \omega) + \iota_X (d\omega) = 0$

Since $d\omega = 0$ , this simplifies to $d(\iota_X \omega) = 0$ , meaning the 1-form $\iota_X \omega$ is closed.

Symplectic vector fields generate one-parameter groups of symplectomorphisms. They are the infinitesimal analogue of symplectomorphisms, just as Killing fields relate to isometries in Riemannian geometry.

DefinitionHamiltonian Vector Field

A vector field $X$ is Hamiltonian if there exists a function $H: M \to \mathbb{R}$ such that:

$\iota_X \omega = dH$

The function $H$ is called the Hamiltonian function and $X$ is denoted $X_H$ . The pair $(M, \omega, H)$ defines a Hamiltonian system.

ExampleHamilton's Equations

In canonical coordinates $(q_i, p_i)$ on $(\mathbb{R}^{2n}, \omega_0)$ , the Hamiltonian vector field $X_H$ satisfies:

$\iota_{X_H} \omega_0 = dH$

Writing $X_H = \sum \dot{q}_i \frac{\partial}{\partial q_i} + \dot{p}_i \frac{\partial}{\partial p_i}$ , this gives:

$\dot{q}_i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q_i}$

These are Hamilton's equations, the fundamental equations of classical mechanics.

Remark

Every Hamiltonian vector field is symplectic, but the converse holds only when $H^1(M, \mathbb{R}) = 0$ . On simply connected manifolds, every symplectic vector field is Hamiltonian. The space of Hamiltonian vector fields modulo locally Hamiltonian ones measures the first de Rham cohomology.

The Hamiltonian $H$ is unique up to addition of constants. If $X_H = X_{H'}$ , then $d(H - H') = 0$ , so $H - H'$ is locally constant. On connected manifolds, $H$ and $H'$ differ by a global constant.

DefinitionPoisson Bracket

The Poisson bracket of functions $f, g \in C^\infty(M)$ is defined by:

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

This makes $C^\infty(M)$ into a Lie algebra with the symplectic form providing the bracket structure.

ExampleCanonical Poisson Brackets

On $(\mathbb{R}^{2n}, \omega_0)$ with coordinates $(q_i, p_i)$ , the Poisson brackets of the coordinate functions are:

$\{q_i, q_j\} = 0, \quad \{p_i, p_j\} = 0, \quad \{q_i, p_j\} = \delta_{ij}$

For general functions $f, g$ :

$\{f, g\} = \sum_{i=1}^n \left( \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} \right)$

The Poisson bracket satisfies the Jacobi identity $\{f, \{g, h\}\} + \{g, \{h, f\}\} + \{h, \{f, g\}\} = 0$ , making $(C^\infty(M), \{\cdot, \cdot\})$ a Poisson algebra. This algebraic structure encodes the entire symplectic geometry.

Remark

Noether's theorem in symplectic language: if $H$ is invariant under the flow of $X_f$ (i.e., $X_f(H) = \{f, H\} = 0$ ), then $f$ is a conserved quantity along Hamiltonian flow. Conservation laws correspond to symmetries via Poisson brackets.

DefinitionHamiltonian Flow

The Hamiltonian flow $\phi_t^H$ of $H$ is the one-parameter group generated by $X_H$ . It satisfies:

$\frac{d}{dt}\phi_t^H = X_H \circ \phi_t^H, \quad \phi_0^H = \text{id}$

Each $\phi_t^H$ is a symplectomorphism, preserving both $\omega$ and $H$ .

ExampleIntegrable Systems

A Hamiltonian system on $M^{2n}$ is completely integrable if there exist $n$ functionally independent functions $f_1 = H, f_2, \ldots, f_n$ in involution:

$\{f_i, f_j\} = 0 \text{ for all } i, j$

The Arnol'd-Liouville theorem states that the level sets are Lagrangian tori, and the motion is quasi-periodic.