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.

Let ${\displaystyle (M,{\mathcal {L}})}$ be a mechanical system with configuration space ${\displaystyle M}$ and smooth Lagrangian ${\displaystyle {\mathcal {L}}.}$ Select a standard coordinate system ${\displaystyle ({\boldsymbol {q}},{\boldsymbol {\dot {q}}})}$ on the tangent bundle ${\displaystyle TM.}$ 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 ${\displaystyle E_{\mathcal {L}}}$ into a function ${\displaystyle {\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}},t)}$ known as the Hamiltonian. The Hamiltonian satisfies $${\displaystyle {\mathcal {H}}\left({\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {\dot {q}}}}},{\boldsymbol {q}},t\right)=E_{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)}$$ which implies that $${\displaystyle {\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}},t)=\sum _{i=1}^{n}p_{i}{\dot {q}}^{i}-{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t),}$$ where the velocities ${\displaystyle {\boldsymbol {\dot {q}}}=({\dot {q}}^{1},\ldots ,{\dot {q}}^{n})}$ are found from the (${\displaystyle n}$-dimensional) equation ${\displaystyle \textstyle {\boldsymbol {p}}={\partial {\mathcal {L}}}/{\partial {\boldsymbol {\dot {q}}}}}$ which, by assumption, is uniquely solvable for ⁠${\displaystyle {\boldsymbol {\dot {q}}}}$⁠. The (${\displaystyle 2n}$-dimensional) pair ${\displaystyle ({\boldsymbol {p}},{\boldsymbol {q}})}$ is called phase space coordinates. (Also canonical coordinates).

In phase space coordinates ⁠${\displaystyle ({\boldsymbol {p}},{\boldsymbol {q}})}$⁠, the (${\displaystyle n}$-dimensional) Euler–Lagrange equation $${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {q}}}}-{\frac {d}{dt}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {\boldsymbol {q}}}}}=0}$$ becomes Hamilton's equations in ${\displaystyle 2n}$ dimensions

${\displaystyle {\frac {\mathrm {d} {\boldsymbol {q}}}{\mathrm {d} t}}={\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {p}}}},\quad {\frac {\mathrm {d} {\boldsymbol {p}}}{\mathrm {d} t}}=-{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}}.}$

The Hamiltonian ${\displaystyle {\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}})}$ is the Legendre transform of the Lagrangian ${\displaystyle {\mathcal {L}}({\boldsymbol {q}},{\dot {\boldsymbol {q}}})}$, thus one has $${\displaystyle {\mathcal {L}}({\boldsymbol {q}},{\dot {\boldsymbol {q}}})+{\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}})={\boldsymbol {p}}{\dot {\boldsymbol {q}}}}$$ and thus $${\displaystyle {\begin{aligned}{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {p}}}}&={\dot {\boldsymbol {q}}}\\{\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {q}}}}&=-{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}},\end{aligned}}}$$

Besides, since ${\displaystyle {\boldsymbol {p}}=\partial {\mathcal {L}}/\partial {\dot {\boldsymbol {q}}}}$, the Euler–Lagrange equations yield ${\displaystyle {\dot {\boldsymbol {p}}}={\frac {\mathrm {d} {\boldsymbol {p}}}{\mathrm {d} t}}={\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {q}}}}=-{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}}.}$