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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.

Overview

Phase space coordinates (p, q) and Hamiltonian H

Let ${\displaystyle (M,{\mathcal {L}})}$ be a mechanical system with the configuration space ${\displaystyle M}$ and the smooth Lagrangian ${\displaystyle {\mathcal {L}}.}$ Select a standard coordinate system ${\displaystyle ({\boldsymbol {q}},{\boldsymbol {\dot {q}}})}$ on ${\displaystyle M.}$ The quantities ${\displaystyle \textstyle p_{i}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)~{\stackrel {\text{def}}{=}}~{\partial {\mathcal {L}}}/{\partial {\dot {q}}^{i}}}$ are called momenta. (Also generalized momenta, conjugate momenta, and canonical momenta). For a time instant ${\displaystyle t,}$ the Legendre transformation of ${\displaystyle {\mathcal {L}}}$ is defined as the map ${\displaystyle ({\boldsymbol {q}},{\boldsymbol {\dot {q}}})\to \left({\boldsymbol {p}},{\boldsymbol {q}}\right)}$ which is assumed to have a smooth inverse ${\displaystyle ({\boldsymbol {p}},{\boldsymbol {q}})\to ({\boldsymbol {q}},{\boldsymbol {\dot {q}}}).}$ For a system with ${\displaystyle n}$ degrees of freedom, the Lagrangian mechanics defines the energy function

$${\displaystyle E_{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)\,{\stackrel {\text{def}}{=}}\,\sum _{i=1}^{n}{\dot {q}}^{i}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}-{\mathcal {L}}.}$$

The Legendre transform of ${\displaystyle {\mathcal {L}}}$ turns ${\displaystyle E_{}}$Zdroj:https://en.wikipedia.org?pojem=Hamiltonian_mechanics
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