I am an electrical engineer and pretty familiar with optimization at the level of books like Boyd & Vandenberghe and Bertsimas' Linear Optimization. I am trying get familiar with the basic concepts of Lagrangian and Hamiltonian mechanics. It's clear there is a lot of overlap here and that these problems are intimately related, but I'm having trouble connecting the dots.
A key issue I have encountered is that a lot of things have some 100 different names, e.g., "convex conjugate = Fenchel transform = Legendre transform, etc" and of course the terminology used in physics is usually not the same as that in the optimization literature that I am familiar with.
Does anyone familiar with both optimization and mechanics have a good source that unifies things nicely? I'm thinking maybe there's some stuff in the older controls literature.
It may be worth a shot to have a look at Gauss principle of least constraint (GPoLC). There is very few material available about it, but I think it is worth the effort to get a hold on it. It basically says that the acceleration that a system experiences is such that the Euclidian norm of the difference between the acceleration it actually experiences and the acceleration it would experience without any constraints is minimised. Consider for example the link of a robot arm. The acceleration "without constraints" would be the acceleration it would experience if you were to cut it loose from any attached links and only be considering gravity, drag etc. but not the reaction forces occurring in the joints.
I recently wanted to implement a rigid body simulation of a bipedal robot with contact constraints. I was very confused by a lot of materials that I found about Lagrangian mechanics and them always being derived in a slightly different way and it was GPoLC that came to rescue.
I think the power of GPoLC is that it turns the problem of finding the equations of motion of a mechanical system into an optimisation problem with quadratic objective. If you add for example time-invariant holonomic constraints to your problem, it is not so hard to see that the feasible set will be convex and hence the problem itself will be convex which allows you to use the full power of convex optimisation results. Constraint forces for example will turn out to be Lagrange multipliers in your problem.