Trajectories never cross in phase-space

[Higgsono]

I heard this statement from time to time, but what does it really mean?

 

[S.G. Janssens]

I heard this statement from time to time, but what does it really mean?

In the context of autonomous ordinary differential equations (ODEs) it means that given any initial condition $x_0$ in phase space there exists a unique trajectory passing through $x_0$.

If different trajectories were to cross at $x_0$, this would imply that the initial-value problem for the ODE would admit multiple solutions on some (possibly small) time interval. When the right-hand side of the ODE admits an appropriate mathematical (Lipschitz) condition, this cannot happen by e.g. the Picard–Lindelöf theorem on the existence of unique local solutions to ODE.

Physically, e.g. in a mechanical context, it means that once the coordinates and momenta at a certain time are known, the future state (i.e. the future coordinates and momenta) of the system is uniquely determined.

 

[Higgsono]

In the context of autonomous ordinary differential equations (ODEs) it means that given any initial condition $x_0$ in phase space there exists a unique trajectory passing through $x_0$.

If different trajectories were to cross at $x_0$, this would imply that the initial-value problem for the ODE would admit multiple solutions on some (possibly small) time interval. When the right-hand side of the ODE admits an appropriate mathematical (Lipschitz) condition, this cannot happen by e.g. the Picard–Lindelöf theorem on the existence of unique local solutions to ODE.

Physically, e.g. in a mechanical context, it means that once the coordinates and momenta at a certain time are known, the future state (i.e. the future coordinates and momenta) of the system is uniquely determined.

Is this the reason that the volume in phase-space for a closed system is conserved? I mean, the only way the volume could change for a closed system would be for the trajectories to cross or merge or split into more then one.

 

[S.G. Janssens]

Is this the reason that the volume in phase-space for a closed system is conserved?

No, volume conservation is due to the specific structure of, say, time-independent Hamiltonian systems. It does not follow from uniqueness of solutions to the initial-value problem alone.