Time-dependent Hamiltonian system - Imported Knowlpedia Content

Definition

Let $I\subseteq\mathbb R$ be an open interval and let $(M,\omega)$ be a symplectic manifold. A time-dependent Hamiltonian system is a smooth function

H:I\times M\to\mathbb R,\qquad H_t(x)=H(t,x),

together with this fixed symplectic phase space. For each $t$ , its Hamiltonian vector field $X_{H_t}$ is defined by

\iota_{X_{H_t}}\omega=d_MH_t.

A trajectory is a curve $\gamma$ satisfying the nonautonomous equation $\dot\gamma(t)=X_{H_t}(\gamma(t))$ . The differential $d_M$ acts only in the $M$ -direction.

Evolution and symplecticity

Local existence and uniqueness produce evolution maps $\Phi_{t,s}$ with $\Phi_{s,s}=\operatorname{id}$ and

\Phi_{t,r}\circ\Phi_{r,s}=\Phi_{t,s}

wherever all terms are defined. Each $\Phi_{t,s}$ is symplectic because every instantaneous field $X_{H_t}$ preserves $\omega$ . In general there is no one-parameter group $\Phi_t$ : the evolution depends on both initial and final time. This distinction is emphasized in Abraham and Marsden, §3.3.

Energy balance and autonomous extension

Along a trajectory,

\frac{d}{dt}H(t,\gamma(t))=\frac{\partial H}{\partial t}(t,\gamma(t)),

because $d_MH_t(X_{H_t})=0$ . Thus explicit time dependence obstructs conservation of the instantaneous Hamiltonian.

The system can be encoded as an autonomous Hamiltonian system on an extended phase space by adjoining time and its conjugate momentum. This construction, including its coordinate equations, is developed in Arnol'd, Chapter 9.

Conventions and scope

References

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989. DOI record. Relevant: Chapter 9, time-dependent Hamiltonian mechanics and extended phase space.
Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, 2nd ed., AMS Chelsea Publishing, 2008. DOI record. Relevant: §3.3, Hamiltonian systems and time-dependent flows.