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Hamilton's equations

These first-order differential equations give the time evolution of generalized coordinates and conjugate momenta of a system from its Hamiltonian function. They are the standard form of classical equations of motion in Hamiltonian mechanics.

\begin{aligned}\frac{\mathrm{d}\boldsymbol{p}}{\mathrm{d}t} &= -\frac{\partial \mathcal{H}}{\partial \boldsymbol{q}} \\ \frac{\mathrm{d}\boldsymbol{q}}{\mathrm{d}t} &= \frac{\partial \mathcal{H}}{\partial \boldsymbol{p}}\end{aligned}

What the symbols mean

generalized coordinates of the system
generalized momenta conjugate to the coordinates
Hamiltonian function of the system, usually the total energy expressed in q and p
time

About this formula

  • FieldPhysics
  • TopicClassical mechanics
  • LevelUniversity
  • Also calledHamilton's equations of motion, Hamiltonian equations of motion, canonical equations of motion, Hamiltonian mechanics equations, momentum and position time derivatives Hamiltonian
  • SourceWikidata Q1115699

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