# 2.5] What is a degree of freedom? (nonlinear science)

## Description

This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by
others.

# 2.5] What is a degree of freedom? (nonlinear science)

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 PoincarĂ©

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

longer conserved).

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