Hamilton's equations - Imported Knowlpedia Content

Statement

Let $(M,\omega,H)$ be an autonomous Hamiltonian system. In Darboux coordinates $(q^1,\ldots,q^n,p_1,\ldots,p_n)$ , where

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

a curve $t\mapsto(q(t),p(t))$ is a Hamiltonian trajectory if and only if it satisfies Hamilton's equations

\dot q^i=\frac{\partial H}{\partial p_i}, \qquad \dot p_i=-\frac{\partial H}{\partial q^i}

for every $i$ . These signs correspond to the convention $\iota_{X_H}\omega=dH$ . The equations give the local coordinate form of the intrinsic vector-field equation $\dot\gamma=X_H$ .

Derivation

Write

X_H=\sum_i\left(a^i\frac{\partial}{\partial q^i} +b_i\frac{\partial}{\partial p_i}\right).

Contracting with the Darboux form gives

\iota_{X_H}\omega=\sum_i(a^i\,dp_i-b_i\,dq^i).

Comparison with $dH=\sum_i(H_{q^i}\,dq^i+H_{p_i}\,dp_i)$ yields $a^i=H_{p_i}$ and $b_i=-H_{q^i}$ . Since trajectories satisfy $\dot\gamma=X_H$ , the coordinate equations follow. See Abraham and Marsden, §3.3.

Examples and time dependence

For $H(q,p)=p^2/(2m)+m\Omega^2q^2/2$ , Hamilton's equations are

\dot q=p/m,\qquad \dot p=-m\Omega^2q,

and hence $\ddot q+\Omega^2q=0$ .

For a time-dependent Hamiltonian $H(t,q,p)$ , the same formulas hold with the partial derivatives evaluated at time $t$ . The resulting evolution is nonautonomous, and $H$ need not be conserved Arnol'd, Chapter 9.

Conventions and scope

References

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989. DOI record. Relevant: Chapters 8–9, Hamilton's equations and time-dependent systems.
Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, 2nd ed., AMS Chelsea Publishing, 2008. DOI record. Relevant: §3.3, Hamiltonian systems in canonical coordinates.