Hamiltonian function - Imported Knowlpedia Content

Definition

Let $(M,\omega)$ be a symplectic manifold. A Hamiltonian function is a smooth function $H:M\to\mathbb{R}$ . With the convention

\iota_{X_H}\omega=dH,

the nondegeneracy of $\omega$ determines a unique vector field $X_H$ , called the Hamiltonian vector field of $H$ . The corresponding Hamiltonian system is the differential equation $\dot{\gamma}(t)=X_H(\gamma(t))$ . Thus $H$ is the scalar function, not the vector field or its flow. Two Hamiltonians differing by a locally constant function generate the same vector field; the converse also holds.

Generated dynamics

Cartan's formula gives $\mathcal{L}_{X_H}\omega=0$ , so every local flow map of $X_H$ preserves the symplectic form. Moreover,

\frac{d}{dt}H(\gamma(t))=dH(X_H)=\omega(X_H,X_H)=0,

and hence $H$ is constant along its own trajectories. These conclusions use only the defining equation and the closedness and skew-symmetry of $\omega$ .

Examples and scope

On $\mathbb{R}^{2n}$ with coordinates $(q_i,p_i)$ and $\omega=\sum_i dq_i\wedge dp_i$ , the defining equation yields

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).

A constant Hamiltonian generates the zero vector field. A vector field preserving $\omega$ need not be globally Hamiltonian: $\iota_X\omega$ must be exact, not merely closed.

Conventions

References

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989. DOI record. Relevant: Chapter 8.
R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed., AMS Chelsea, 2008. DOI record. Relevant: §3.3.