First Integral

A first integral of Time Evolution of Dynamical Variables is a non-constant solution s.t.

Each first integral reduces the order of this equation. In particular, if is a first integral with , then the trajectory will be confined to a hypersurface If we find independent first integrals, the trajectory will be in their intersection, which in general has dimension . If , then the trajectory is completely determined (it is just a curve).

Unfortunately, in general the surfaces are degenerate and do not separate the points so we can’t rly do this. Sufficiently complicated systems behave in a more probabilistic manner (think thermodynamics), but it is a millennium problem to see if this follows from our equations (ergodic hypothesis)

In this course we only consider sufficiently nice equations.