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A Hamiltonian system with d degrees of freedom satisfies Hamilton's
equations of motion [2]:
and represent the position and momentum
respectively and are d-dimensional vectors. is the
gradient operator taken w.r.t. , denotes the
derivative of with respect to time, t. is the scalar-valued, autonomous (time-independent)
Hamiltonian function. A Hamiltonian system is thus (necessarily) of even dimension
(with m=2d in the autonomous form of (1)).
The Hamiltonian is time-invariant, i.e. it is a constant of the motion.
When the Hamiltonian is interpreted as the energy of the system, time-invariance
is equivalent to conservation of energy. To see this consider the chain
rule:
Jorge Romao
5/14/1998