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Note: in this question when I talk about "phase space," I will be refering to velocity vs. position space, which can also be correctly referred to as "state space." Many sources (including John R. Taylor's Classical Mechanics) allow for both terminologies. This discussion may also apply when the velocity is replaced by the generalized momentum... but I'm specifically interested in velocity vs. position.

In phase space, you will commonly hear the refrain that trajectories cannot cross. This is, of course, true with a simple harmonic oscillator (ellipses forever), a damped harmonic oscillator (an inward elliptical spiral), and more. However, when dealing with some nonlinear oscillators (and certainly when dealing with chaos) the trajectories DO cross regularly. For example, a driven damped pendulum (DDP) clearly has crossing trajectories.

Chaos, however, is not a sufficient criterion for crossing trajectories. A period-2 DDP (i.e. one that flips back and forth between two orbits in a periodic fashion) crosses its own trajectory twice per cycle. A period-4 DDP does it 4 times!

What is the criterion (or criteria) for when a phase space trajectory may cross?

related, but not particularly helpful to me: When can phase trajectories cross?

Qmechanic
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Bunji
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3 Answers3

17

The general idea is that trajectories that cross on phase space will have identical initial conditions from that point and will therefore satisfy identical initial-value problems, and from that you can conclude that they will identically track each other.

The example you give is of a driven system, which is subject to an external driving force that depends on time and is therefore not time-translation invariant. For such a system, trajectories can cross at different times in the driving cycle, but you cannot conclude that they are solutions of identical IVPs, because the initial conditions are the same but the driving is not.

If you restrict your attention to systems that are invariant under time translations, this feature will go away.

Emilio Pisanty
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  1. Although Hamiltonian systems and symplectic phase spaces have nicer properties, it is usually not important when we discuss whether trajectories are forbidden to cross. Hence it is fine that OP uses velocities rather than momenta.

  2. It is much more important that the governing ODE (apart from mild regularity requirements) is

    • 1st order

    and

    cf. e.g. my Phys.SE answer here.

  3. In particular, OP's example with a driven pendulum is not autonomous.

Qmechanic
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2

When you have a phase space where lines can cross, you have forgotten something. Namely some parameter that influences future system behavior, which you have forgotten to include into your phase space.

For the system in the example, that parameter is the phase of the driving force. I.e. the phase space is 3D, not 2D. If you include that third dimension, your crossings will simply disappear.

cmaster - reinstate monica
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