Non intersecting phase space trajectories

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In an autonomous system of equations, phase space trajectories do not intersect, which can be proven mathematically. If two trajectories were to intersect, it would imply that the system could have two different future states from the same initial condition, violating the uniqueness of solutions. The physical significance of non-intersecting trajectories indicates that the system's evolution is deterministic and predictable. If trajectories were to intersect, it could lead to chaotic behavior or multiple outcomes from a single state, complicating the system's dynamics. Thus, the non-intersection of trajectories is crucial for maintaining the integrity of the system's predictive nature.
geet89
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The phase space trajectories of an autonomous system of equations don't intersect.

Can this be proved mathematically.

Also what is the physical significance of this statement. What happens if they intersect?
 
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geet89 said:
The phase space trajectories of an autonomous system of equations don't intersect.

Can this be proved mathematically.

Also what is the physical significance of this statement. What happens if they intersect?

Hint: What would happen if you had two trajectories departing from the same point (which is another way of saying that there are intersecting trajectories)?
 

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