Hamiltonian mechanics

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${\displaystyle {\textbf {F}}={\frac {d}{dt}}(m{\textbf {v}})}$ Second law of motion
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In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton,^[1] 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.