Graphically, is each line segment supposed to be tangent to the circle at that radius?

john-b-yang

@RichardChen9 I'm not quite sure either but I don't believe so? I feel like if a line was tangent to any circle, it should, in a sense, be parallel to all lines tangent to all the other circles at that same point, if that makes sense? I think the lines are just meant to represent the aggregate effect of spiraling out, aka the combined result of having springs that are not moving in a symmetric manner to produce a more controlled movement. Would be great if a TA could confirm!

yzyz

Yes each line segment is supposed to be tangent to the circle at that radius. In the ideal world, the motion of the spring will trace out a circle in the phase diagram. In our discretized simulation, we calculate the direction we are moving in at one timestep and move in that direction until the next timestep, which shows up as a line segment in the diagram. At any point in time the direction of our movement is tangent to the circle, which is why each line segment is tangent to its circle.

lhagaman

A fun fact about phase space is that for an ensemble of starting states occupying a region in phase space, the area of that phase space is conserved as the system evolves. It is obvious in this case since the starting area will just spin around the origin, but it is true in general for much more complex systems. See https://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian)

Graphically, is each line segment supposed to be tangent to the circle at that radius?

@RichardChen9 I'm not quite sure either but I don't believe so? I feel like if a line was tangent to any circle, it should, in a sense, be parallel to all lines tangent to all the other circles at that same point, if that makes sense? I think the lines are just meant to represent the aggregate effect of spiraling out, aka the combined result of having springs that are not moving in a symmetric manner to produce a more controlled movement. Would be great if a TA could confirm!

Yes each line segment is supposed to be tangent to the circle at that radius. In the ideal world, the motion of the spring will trace out a circle in the phase diagram. In our discretized simulation, we calculate the direction we are moving in at one timestep and move in that direction until the next timestep, which shows up as a line segment in the diagram. At any point in time the direction of our movement is tangent to the circle, which is why each line segment is tangent to its circle.

A fun fact about phase space is that for an ensemble of starting states occupying a region in phase space, the area of that phase space is conserved as the system evolves. It is obvious in this case since the starting area will just spin around the origin, but it is true in general for much more complex systems. See https://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian)