|
|
|
|
Hamilton equations
|
(Definition)
|
|
|
The Hamilton equations are a formulation of the equations of motion in classical mechanics.
Suppose
 is an open set, suppose $I$ is an interval (representing time), and
 is a smooth function. Then the equations
are the Hamilton equations for the curve
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.
Suppose $P$ is a symplectic manifold with symplectic form $\omega$ and that
 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 (![[*]](http://images.planetmath.org:8080/cache/objects/6410/js//usr/share/latex2html/icons/crossref.png "[*]") )-().
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
 is known as state space.
|
"Hamilton equations" is owned by CWoo. [ full author list (4) | owner history (3) ]
|
|
(view preamble | get metadata)
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
There are 8 references to this entry.
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 4865 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) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
