Time-reversal symmetry

If (r₁(t),...,r_3N(t),p₁(t),...,p_3N(t)) is a solution
then (r₁(-t),...,r_3N(-t),-p₁(t),...,-p_3N(t)) is also a solution.

This shows that the equation of motion has no preferred direction for times arrow. If we watch a movie of a system evolving under a Hamiltonian equation of motion, it is impossible to decide if we are watching the movie in its original form or if it is being shown in reverse.