Simulate the dynamics of a 3D discrete rigid body from "first principles"

Question

The question stems from wanting to implement a rigid body dynamics simulation without resorting to any of the well-known concepts that are commonly used, such as moment of inertia, torque, and angular velocity. I only want to use f=ma on point masses, together with some other notion of force transfer that I haven't identified yet.

Here is my model of the problem. A massless support structure (light blue), with some small heavy spheres (black) embedded. The spheres are precisely embedded in the structure and they are not allowed to move around in it, although they can revolve on themselves; there is no friction between the spheres and the support structure. The structure is only intended as a conceptual device to transmit forces; it delivers external forces such as f to the spheres, and mutual internal forces between spheres, that are responsible for keeping them at constant distances to each other.

The problem Let's assume that we know each mass m[i], the external force f with its point of application q, and all the state variables in the system, that is, I think, all the positions p[i] and velocities v[i] for each point-like mass. We want to compute all accelerations a[i].

The questions

• Is the problem above well-defined?
• In other words, is the acceleration a[i] a function of all the other variables mentioned above?
• Is this function analytic?

Answer 1

You can do it, and it will be well defined. The key would be adding the constraints for a rigid body: the distances between masses never changes.

That being said, what you are doing is really inventing the exact mathematical problem that leads to things like moments of inertia. If successful, you will have found a way to simulate something in 1,000 equations that we know how to simulate in 4.

If you are interested in first principles, may I recommend delving into group theory. Once you understand that the rigid body assumptions define a group E(3), you will see that the system you created can only be the same as the usual set of equations

However, if you toss out the rigid body assumptions, and instead do something like springs between masses, you get into finite element analysis, which is a really strange interesting thing.

Answer 2

As stated in @CortAmmon's answer, the problem can be resolved as you suggest, provided the correct number of independent rigidity constraints is added to the external forces.

I would not be negative about that idea, as in the mentioned answer, because in some cases, the numerical solution of the constrained equations of motion can be preferred over the integration of the rigid body equation for motion around the center of mass.

Algorithms for numerical integration of the equations of motion in the presence of rigidity constraints have been developed over the years. However, a word of caution is necessary to avoid a naive implementation. In particular, it is important to be aware that evaluating the constraint reaction forces (for example, through the Lagrange multiplier technique) is not enough. If we use them in the numerical integration algorithm, the constraint would be satisfied only within its accuracy. For example, using a symplectic Verlet algorithm, we would get the rigidity constraint satisfied only with a global error proportional to the square of the time step. That is not acceptable in the majority of the cases.

Special algorithms dependent on the integration algorithm have been introduced since the pioneering work by Ciccotti and Ryckaert, as summarized in the review paper G. Ciccotti, and J.-P. Ryckaert. "Molecular dynamics simulation of rigid molecules." Computer Physics Reports 4 (1986): 346-392. The basic idea behind such algorithms is to satisfy the constraint with the maximum accuracy at each time step, introducing an error in the constraint forces that is not larger than the integration algorithm error. The cited paper provides more details.