# Hamilton equations

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

## Local formulation

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

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

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

are the *Hamilton equations* for the curve

$$(q,p)=({q}_{1},\mathrm{\dots},{q}_{n},{p}_{1},\mathrm{\dots},{p}_{n}):I\to U\times {\mathbb{R}}^{n}.$$ |

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 $H:P\to \mathbb{R}$
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 (http://planetmath.org/Poincare1Form). (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 ${x}^{1},\mathrm{\dots}{x}^{2n}$ on the manifold $P$, they can be written as follows:

$${\dot{x}}^{i}={({X}_{H})}^{i}({x}_{1},\mathrm{\dots}{x}_{2n},t)$$ |

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^{} $P\times \mathbb{R}$ is known as state space.

Title | Hamilton equations |
---|---|

Canonical name | HamiltonEquations |

Date of creation | 2013-03-22 14:45:58 |

Last modified on | 2013-03-22 14:45:58 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 9 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 53D05 |

Classification | msc 70H05 |

Related topic | Quantization |