Why Can't Flows in Phase Space Cross?

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SUMMARY

Flows in phase space cannot cross due to the deterministic nature of dynamical systems, which ensures that identical initial conditions lead to unique trajectories. This principle is upheld in time-invariant systems, where Liouville's Theorem states that volumes in phase space are preserved. When considering Hamiltonian systems with explicit time dependence, the phase space can be extended to include time and energy as additional dimensions, reinforcing that flows will not intersect unless the system is non-deterministic.

PREREQUISITES
  • Understanding of Hamiltonian mechanics
  • Familiarity with Liouville's Theorem
  • Knowledge of phase space concepts
  • Basic principles of deterministic systems
NEXT STEPS
  • Study Hamiltonian mechanics and its implications on phase space
  • Explore Liouville's Theorem in detail
  • Learn about deterministic vs. non-deterministic systems
  • Investigate the extension of phase space in systems with explicit time dependence
USEFUL FOR

Physicists, mathematicians, and students studying dynamical systems, particularly those interested in the behavior of phase space and the implications of determinism in mechanics.

Eidos
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Hello ladies and gentlemen

Why can't flows in phase space cross?
Would it imply that the system may be at the same state at some future time and then follow a different trajectory? That is to say that the identical initial condition gives a different final condition.

To my mind, flows in phase space would only not cross if the system is time invariant.

Slightly related, non-dissipative systems have their volumes preserved in phase space (Liouville's Theorem), is that the total volume of the phase space or any selectable portion of it?

Thanks for any replies :smile:
 
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If the Hamiltonian has explicit time dependence, you can always extend the phase space to include t and E as extra dimensions (in fact, in relativity, one should always include these - conservation of energy will, then, define a surface in the phase space which the system is constrained to remain on). With this extension, it should be clear that even without time invariance, the phase space flows will not cross unless the system is not deterministic.
 
Cool thanks that clears up a number of things that have been troubling me.

:smile:
 

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