Visualizations

These are some .gif animations I generated in Mathematica modeling a double pendulum system, comprised of two equal point masses connected to each other and an axis of rotation by massless rigid rods of equal length. Note that for both animations I chose a time step size such that one second of motion was calculated using 2000 steps. However, to generate the animations I only plotted the positions of each 10th step in order to keep the size of the .gif files reasonable.

Discrete Euler-Lagrange Model

This animation was generated using an update map derived from the discrete Euler-Lagrange equations. The configuration coordinates are the angles of displacement from the vertical axis for each rod/mass pendulum, so this is a 2-dimensional system. The frame used to describe velocity components is the coordinate-induced frame.

While not exactly conserved, the energy is bounded within a very reasonable range.
Double pendulum modeled with discrete Euler-Lagrange formula

Notice that the total energy of the system, while not conserved, is bounded. In the physical world, the real (but still idealized to exclude nonconservative forces)  system should conserve energy, that is the energy should be constant. However, having bounded energy is a good trade-off in modeling because we can expect reasonable long-term accuracy in our simulation.

Discrete Hamel Model

I’ve also included an animation of the system generated by a preliminary algorithm that I’ve developed for Hamel’s equations. Recall that the Euler-Lagrange equations are a special case of Hamel’s equations where our choice of frame happens to coincide with the frame induced by our choice of configuration coordinates. For the model below I’ve chosen a different frame, one that diagonalizes the mass tensor in the Lagrangian. Unfortunately, there is not a nice physical description of this choice of frame; its result is simply that it removes the coupling of velocity components in the kinetic energy.

Again, energy is bounded. The difference between the behavior of energy between the two models is striking.

Double pendulum modeled with discrete Hamel formula

The overall behavior of the system modeled with this choice of frame is similar to the behavior of the system generated by the discrete Euler-Lagrange equations. The energy, as expected, is still bounded. However, the choice of frame induces different energy behavior within the energy bounds. I find this intriguing, and I’m currently examining possible causes for this change in energy behavior and potential advantages. Perhaps this can provide some insight into how to systematically choose non-coordinate frames for mechanical systems.

Advertisements