This article is from the Nonlinear Science FAQ, by James D. Meiss email@example.com with numerous contributions by others.
The notion of "degrees of freedom" as it is used for
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Hamiltonian systems means
one canonical conjugate pair, a configuration, q, and its conjugate momentum
p. Hamiltonian systems (sometimes mistakenly identified with the notion of
conservative systems) always have such pairs of variables, and so the phase
space is even dimensional.
In the study of dissipative systems the term "degree of freedom" is often used
differently, to mean a single coordinate dimension of the phase space. This
can lead to confusion, and it is advisable to check which meaning of the term
is intended in a particular context.
Those with a physics background generally prefer to stick with the Hamiltonian
definition of the term "degree of freedom." For a more general system the
proper term is "order" which is equal to the dimension of the phase space.
Note that a dynamical system with N d.o.f. Hamiltonian nominally moves in a
2N dimensional phase space. However, if H(q,p) is time independent, then
energy is conserved, and therefore the motion is really on a 2N-1 dimensional
energy surface, H(q,p) = E. Thus e.g. the planar, circular restricted 3 body
problem is 2 d.o.f., and motion is on the 3D energy surface of constant
"Jacobi constant." It can be reduced to a 2D area preserving map by Poincaré
section (see [2.6]).
If the Hamiltonian is time dependent, then we generally say it has an
additional 1/2 degree of freedom, since this adds one dimension to the phase
space. (i.e. 1 1/2 d.o.f. means three variables, q, p and t, and energy is no