Hamilton–Jacobi equation in the context of Hamilton's optico-mechanical analogy

In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities ${\displaystyle {\dot {q}}^{i}}$ used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena.

Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics.

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (q, p) → (Q, P) that preserves the form of Hamilton's equations. This is sometimes known as form invariance. Although Hamilton's equations are preserved, it need not preserve the explicit form of the Hamiltonian itself. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi equations (a useful method for calculating conserved quantities) and Liouville's theorem (itself the basis for classical statistical mechanics).

Since Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates q → Q do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if the momentum is simultaneously changed by a Legendre transformation into${\displaystyle P_{i}={\frac {\partial L}{\partial {\dot {Q}}_{i}}}\ ,}$where ${\displaystyle \left\{\ (P_{1},Q_{1}),\ (P_{2},Q_{2}),\ (P_{3},Q_{3}),\ \ldots \ \right\}}$ are the new co‑ordinates, grouped in canonical conjugate pairs of momenta ${\displaystyle P_{i}}$ and corresponding positions ${\displaystyle Q_{i},}$ for ${\displaystyle i=1,2,\ldots \ N,}$ with ${\displaystyle N}$ being the number of degrees of freedom in both co‑ordinate systems.