nLab Hamilton's equations

Idea

The equation of motion as expressed in Hamiltonian mechanics.

Given a phase space represented by a symplectic manifold $(X,\omega)$, and given a Hamiltonian function $H \colon X \longrightarrow \mathbb{R}$ the solutions to the equations of motion are trajectories $\gamma \colon \mathbb{R} \longrightarrow X$ which satisfy

$$\mathbf{d} H(-) = \omega(\dot \gamma,-) \,,$$

hence which are flow lines of the flow induced by the Hamiltonian vector field associated with $H$.

Related concepts

• classical mechanics

  • prequantized Lagrangian correspondence

• differential equation

• equations of motion

  • Newton's laws of motion

  • Hamilton’s equations, de Donder-Weyl-Hamilton equation

    • symplectic integrator

  • Einstein's equations

• Liouville's equation

References

• Richard Fitzpatrick, Newtonian Dynamics – Hamilton’s equations