- #1

Cleonis

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- TL;DR Summary
- Looking for a trajectory integrator that also supports cubic potential, in order to corroborate a trajectory found with an interactive diagram that implements in numerical analysis form Hamilton's stationary action.

The context:

I created an educational resource, a set of interactive diagrams that allow the user to see how Hamilton's stationary action arrrives at the true trajectory. There is a diagram for each of the following three cases:

- Uniform force, hence the potential increases linear with position

- Hooke's law, hence the potential increases quadratic with displacement

- Force increases quadratic with displacement, hence the potential increases with the cube of the displacement.

In each of the three instances the start point is t=-1, height zero, the end point is t=1, height zero.

The total time of 2 units of time is divided in 20 sub-sections, the nodes can be moved with sliders. The sliders modify the ascending part of the trajectory, the descending part is a mirror of the ascending part.

The screenshots below are excepts of the full diagrams. The full diagrams have 4 subpanels. The subpanels not shown here are the subpanel that contains the controlling sliders, and the subpanel displaying the value of Hamilton's action.

As we know, with a linear potential the resulting trajectory is a parabola, and with Hooke's law the resulting motion is harmonic oscillation. The diagrams show that indeed those trajectories correspond to stationary action.

In the case of a potential that increases with the

I would like to show independent corroboration of that trajectory.

I'm confident that the interactive diagram is good.

The red and green dots in the righthand subpanel show the energy.

(Kinetic energy: red dots. The curve for the potential energy has been flipped upside down; the green dots represent the

My question:

I have been looking whether there is an online trajectory integrator that will take

Of course, I can implement such an integrator myself, but for corroboration: the more independent the more convincing.

Does anyone know such an integrator is available somewhere?

I created an educational resource, a set of interactive diagrams that allow the user to see how Hamilton's stationary action arrrives at the true trajectory. There is a diagram for each of the following three cases:

- Uniform force, hence the potential increases linear with position

- Hooke's law, hence the potential increases quadratic with displacement

- Force increases quadratic with displacement, hence the potential increases with the cube of the displacement.

In each of the three instances the start point is t=-1, height zero, the end point is t=1, height zero.

The total time of 2 units of time is divided in 20 sub-sections, the nodes can be moved with sliders. The sliders modify the ascending part of the trajectory, the descending part is a mirror of the ascending part.

The screenshots below are excepts of the full diagrams. The full diagrams have 4 subpanels. The subpanels not shown here are the subpanel that contains the controlling sliders, and the subpanel displaying the value of Hamilton's action.

As we know, with a linear potential the resulting trajectory is a parabola, and with Hooke's law the resulting motion is harmonic oscillation. The diagrams show that indeed those trajectories correspond to stationary action.

In the case of a potential that increases with the

*cube*of the displacement there is (to my knowledge) no analytic solution. So in that case I used the interactive diagram to*find*the corresponding trajectory.I would like to show independent corroboration of that trajectory.

I'm confident that the interactive diagram is good.

The red and green dots in the righthand subpanel show the energy.

(Kinetic energy: red dots. The curve for the potential energy has been flipped upside down; the green dots represent the

*minus*potential energy. With the potential energy curve upside down there is a visual verification. The true trajectory has the property that at every point in time the rate of change of kinetic energy matches the rate of change of potential energy. With the curve for the potential energy upside down the red and green curve are parallel to each other all the time.)My question:

I have been looking whether there is an online trajectory integrator that will take

*any potential*as input, so that I can input a potential that increases with the cube of displacement. All the online trajectory integrators I looked at are for one case only: Earth gravity.Of course, I can implement such an integrator myself, but for corroboration: the more independent the more convincing.

Does anyone know such an integrator is available somewhere?