Hamilton's equations

Hamilton's equations replace the second-order Euler-Lagrange equations with a first-order pair: q̇ᵢ = ∂H/∂pᵢ, ṗᵢ = −∂H/∂qᵢ. Evolution of the system is a flow on 2N-dimensional phase space, and the flow is strictly Hamiltonian in a geometric sense: it preserves the symplectic form dq ∧ dp.

The symmetry between position and momentum that the equations exhibit is the reason phase space is the natural arena for classical mechanics. Canonical transformations that mix q and p — rotations, reflections, generating-function transformations — leave the form of Hamilton's equations invariant.

Derived by William Rowan Hamilton in 1833 as a Legendre transform of the Lagrangian formulation.