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

[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?

Thanks in advance, alk

 

[Nate810]

is. Yes, you are correct to think that the dimension of a dynamical system should be an even number in order for the conversion from higher-order equations to lower-order equations to be possible. This is because of the fact that for every degree of freedom (DOF) in a system, two equations are required (one for the position and one for the velocity). Thus, if a dynamical system is composed of n DOFs, then there must be 2n equations in the system in order to describe it completely. In addition, if the system can be expressed in terms of Lagrangian or Hamiltonian mechanics, then these equations must also satisfy certain energy conservation requirements.