Systems with Two Degrees of Freedom
A system with two degrees of freedom is a system defined by
where is a vector field on .
Conservative System
A system with two degrees of freedom is conservative if there exists a function
such that .
Remark
In a system of one degree of freedom, it is always possible to construct the potential energy by taking the integral, but in a system of two degrees of freedom, the this is not so.
Law of Conservation of Energy
The total energy of a conservative system is conserved, that is,
is a constant along the trajectory of the system.
Proof