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# Hamilton equations

The Hamilton equations are a formulation of the equations of motion in classical mechanics.

# Local formulation

Suppose $U\subseteq\mathbbm{R}^{n}$ is an open set, suppose $I$ is an interval (representing time), and $H\colon U\times\mathbbm{R}^{n}\times I\to\mathbbm{R}$ is a smooth function. Then the equations

$\displaystyle\dot{q}_{j}$ | $\displaystyle=\frac{\partial H}{\partial p_{j}}(q(t),p(t),t),$ | (1) | ||

$\displaystyle\dot{p}_{j}$ | $\displaystyle=-\frac{\partial H}{\partial q_{j}}(q(t),p(t),t),$ | (2) |

are the *Hamilton equations* for the curve

$(q,p)=(q_{1},\ldots,q_{n},p_{1},\ldots,p_{n})\colon I\to U\times\mathbbm{R}^{n}.$ |

Such a solution is called a *bicharacteristic*, and $H$ is
called a *Hamiltonian function*. Here we use classical notation;
the $q_{i}$’s represent the location of the particles,
the $p_{i}$’s represent the momenta of the particles.

# Global formulation

Suppose $P$ is a symplectic manifold with symplectic form $\omega$ and that $H\colon P\to\mathbbm{R}$ is a smooth function. Then $X_{H}$, the Hamiltonian vector field corresponding to $H$ is determined by

$dH=\omega(X_{H},\cdot).$ |

The most common case is when $P$ is the cotangent bundle of a manifold $Q$ equipped with the canonical symplectic form $\omega=-d\alpha$, where $\alpha$ is the Poincaré $1$-form. (Note that other authors may have different sign convention.) Then Hamilton’s equations are the equations for the flow of the vector field $X_{H}$. Given a system of coordinates $x^{1},\ldots x^{{2n}}$ on the manifold $P$, they can be written as follows:

$\dot{x}^{i}=(X_{H})^{i}(x_{1},\ldots x_{{2n}},t)$ |

The relation with the former definition is that in canonical local coordinates $(q_{i},p_{j})$ for $T^{\ast}Q$, the flow of $X_{H}$ is determined by equations (1)-(2).

Also, the following terminology is frequently encountered — the manifold $P$ is known as the phase space, the manifold $Q$ is known as the configuration space, and the product $P\times\mathbbm{R}$ is known as state space.

## Mathematics Subject Classification

53D05*no label found*70H05

*no label found*

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## Corrections

general case by rspuzio ✓

a few small things by pbruin ✓

Classification by invisiblerhino ✓