Star-shaped phase space

I am asked to classify the following phase spaces. The phase spaces 2 and 3 are fairly simple (harmonic oscillator and a elastically reflected particle). However, I fail to classify the phase space number 1.

I'm thinking phase space 1 is impossible since it asymptotically approaches a certain x value, which is impossible for speeds that get lower and lower as the point is approached.

Can anyone point me in some direction to find the physical systems that create the phase space? • The name of the shape is an astroid if that helps, but I also can't think of what it represents. Feb 24 '19 at 23:06
• Graphs 1,2,3 correspond to respective $L_1$, $L_2$ and $L_{\infty}$ norms in this space, if this lights a bulb. In this head it doesn't, sorry.
– kkm
Feb 25 '19 at 0:33

I assume that by classification, you are asked to comment on the broad nature of the potentials producing these trajectories. Indeed, 2 and 3 are for the harmonic potential and the square well potential. Let us use $$\dot x= p$$ for simplicity. You might be asked to take a wild stab at the parity and x-p-symmetric plane curves for the conserved (constant) energy E depicted here, which specifies the Hamiltonian, ipso facto.

I'll ignore 4, since you appear comfortable with it―it is reminiscent of the double-well potential $$(x^2-1)^2$$ trajectories, except it is parity and coordinate-momentum symmetric. It must have a name I don't know (Vinnikov quartic?).

Trajectory 1 is just the astroid, $$E=p^{2/3}+x^{2/3},$$ with all four combinations of signs for real x,p as they go past their respective zeros.

The freaky deceleration cusp around x=0 at the top of your figure is represented by $$\frac{dp}{dx}= -\sqrt{p/x} ~,$$ and you may similarly investigate the other three, mutatis mutandis.

This is a wildly idealized problem, so I don't imagine you are asking for realistic physical realizations of it, but there must be something around... there always is.

NB I don't understand why you might think asymptotic approach to a point is "impossible". Think of a ball rolling towards a hilltop with "just enough" energy. You are trying to integrate the ODE, $$\frac{dx}{dt}= (1-x^{2/3})^{3/2},$$ achievable by $$dt=dx ~ (1-x^{2/3})^{-3/2}$$.

I don't disagree with @CosmasZachos but here's a complementary observation. (Thanks to @kkm for his nice comment.)

For a time dependent system trajectories in phase space are curves of constant energy $$E$$. So each case 1, 2, 3, 4 should be given by some function of $$x,\dot x$$ such that the locus of points where $$E=f(x,\dot x)$$ for fixed $$E$$ is given by said curve (for a different $$f$$ for each curve). Given $$f$$ you can then identify the Hamiltonian, or try to.

As @kkm suggests cases 1-3 arise from $$E=|\dot x|^p + |x|^p$$ (Since $$E^{1/p}$$ is also constant along the trajectory, we see $$E^{1/p}$$ is a curve of constant $$L_p$$-norm.) Specifically curve 1 corresponds to some $$0, curve 2 looks like $$p=2$$, and curve 3 is the limit $$p\to \infty$$*. You can read off the corresponding Hamiltonians. They don't look very physical, except if $$p=2$$.

Curve 4 is not in this class but @CosmasZachos identified it anyway so I will not be commenting on it.

As I commented above I think this problem isn't especially illuminating...

*: You need to replace $$E\to E^{1/p}$$ for the expression to make sense as $$p\to \infty$$