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Hamilton equations (Definition)

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
$\displaystyle (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
$\displaystyle 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:
$\displaystyle \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.



"Hamilton equations" is owned by CWoo. [ full author list (4) | owner history (3) ]
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Cross-references: state space, product, configuration space, local coordinates, relation, coordinates, vector field, flow, canonical, manifold, cotangent bundle, Hamiltonian vector field, symplectic form, symplectic manifold, represent, function, Hamiltonian, solution, curve, smooth function, interval, open set, equations
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This is version 6 of Hamilton equations, born on 2004-10-24, modified 2008-05-13.
Object id is 6410, canonical name is HamiltonianEquations.
Accessed 3704 times total.

Classification:
AMS MSC: 53D05 (Differential geometry :: Symplectic geometry, contact geometry :: Symplectic manifolds, general)
 70H05 (Mechanics of particles and systems :: Hamiltonian and Lagrangian mechanics :: Hamilton's equations)
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