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A Hamiltonian system with 
degrees of freedom satisfies Hamilton's
equations of motion [2]:
and 
represent the position and momentum
respectively and are -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 
in the autonomous form of (
)).
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:
