# Degrees of freedom and conversion of a system of O.D.E.'s into 1 O.D.E.

1. Sep 18, 2011

### alk

I am wrighting a short introduction to dynamical systems, and I can't seem to understand when are we allowed to talk about the "degrees of freedom" of a dynamical system.

A system of one degree of freedom can be described by a 2nd order ordinary differential equation of the form

$\ddot{x}$=f(x,t)______(1) (mechanics $\rightarrow$ Newton's 2nd law)

and this equation can be converted into a system of two O.D.E.'s of the form

$\dot{x}_{1}$=$x_{2}$__________(2a)
$\dot{x}_{2}$=f($x_{1}$,t)_______(2b)

I'm thinking that if and when this conversion is invertible, then starting from eq's (2) as the description of a dynamical system of 2 dimensions, one can say that this dynamical system has one degree of freedom.

The trouble is that I don't know if I'm correct, and I can't find the theorem (if there exists one) that states the conditions under which the conversion of a system of O.D.E.'s into one (or even more) O.D.E. of higher order (lets say 2nd), is possible.

More specifically, Langrangian and Hamiltonian mechanics implies that the dimension of a dynamical system should be an even number. Am I right to think that this may be one of the conditions?