Hamilton's Equations

The equations defined by

$$\dot q = {\partial H\over\partial p}$$ (1)

$$\dot p = -{\partial H\over\partial q},$$ (2)

where $\dot x\equiv dx/dt$ and $H$ is the so-called Hamiltonian, are called Hamilton's equations. These equations frequently arise in problems of celestial mechanics. Another formulation related to Hamilton's equation is

$$p={\partial L\over\partial\dot q},$$ (3)

where $L$ is the so-called Lagrangian.

References

Morse, P. M. and Feshbach, H. ``Hamilton's Principle and Classical Dynamics.'' §3.2 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 280-301, 1953.