A first integral of a system of ordinary differential equations 
is a nonconstant differentiable function 
that is constant along every solution. On a domain where
the solutions and derivatives exist, this condition is expressed by
 |
The chain rule then gives 
. In mechanics, such a conserved quantity is
also called a constant of motion. A time-independent first integral restricts each
trajectory to a level set of 
.
For example, the system 
,
has the first integral 
, since
 |
The nonzero trajectories therefore lie on circles. Continuous variational symmetries provide an important source of first integrals through Noether's symmetry theorem.
