Question

In Hamiltonian Mechanics in the phase space specifically, why is it not possible for two orbits to intersect?

Does the theorem state that the two orbits can't intersect at the same time or at all times ? one way I imagine that is if I have 2 1D objects subject to a net force = 0, if both have the exact same momenta and different positions, as the two systems evolve with time, wouldn't the two orbits at some point intersect (not at the same time of course)? If not then please explain why and explain what the orbit in the phase space represent.

Answer 1

A phase space diagram is a plot of generalized coordinate and its generalized momenta.

An orbit in the phase space is defined as the accumulation of points which are related to each other by the evolution function of the dynamical system.

Now, coming to the question: whether two orbits can intersect each other? Answer is NO

A trajectory in a phase space is determined uniquely by a set of phase space coordinates (generalized coordinate and momenta) . As these points are related to each other by the evolution of time, no two coordinates are equal at different instances of time.