# GSoC blog post

Penultimate week of the GSoC period and it has been busy busy busy.

In my last post I had spoken about opening a PR for LagrangesMethod and about cleaning up the PR on energy functions. (links to both are in last week’s post and I won’t repeat them here.) Much of this week was spent cleaning up both those PRs and quite extensive testing on LagrangesMethod. The testing has been mostly successful. I shall explaing why ‘mostly’ in the a bit. The PR for the renergy functions has been merged and I’m just waiting for approval from ‘the boss’ so that LagrangesMethod can be merged too.

I would like to direct the reader to my proposal. In it I had said that I would write this class only for unconstrained systems. The idea was to modify this to be a ‘complete’ class post the GSoC period. But as we got into the week to begin working on this class, Gilbert and I decided that we would make it a full fledged Lagrange class; that could handle any kind of constraint on mechanical systems. Constraints on mechanical systems are basically of 2 types – configuration constraints (or holonomic constraints) and velocity constraints (or non-holonomic constraints). Depending on the methods used (Newton-Euler or Lagrange or Kane’s method and so on) these constraint equations are accounted for differently In the case of Lagrange’s method, there are additional terms due to these constraints that result in the introduction of the Lagrange multipliers. So, basically, repeating myself for the sake of clairty, one can now obtain the equations of motion in sympy.physics.mechanics using LagrangesMethod for any kind of system. I would even like to go out on a limb (quite literally under my current circumstances :P) and claim that it could be made use of for more generic applications involving ‘The Method of Lagrange Multipliers’ provided the user has the Lagrangian and constraint equations. (The documentation will however be limited to the domain of mechanical systems but shouldn’t be too hard to translate into something more generic for a user). The class also handles nonconservative forces thus making it a complete class.

In terms of testing each of these functionalities, I feel the tests are pretty thorough. I have tested the nonconservative force handling of the class on a simple spring-mass-damper system. I have tested the handling of the constraint equations using the famous ‘disc on an inclined plane’ test problem. Tests on more complex 3d systems have been performed like the rolling disc (more on this later). And then tests on a multibody system have been verified for a double pendulum. All of these work correctly; results have been compared to solutions by ahnd.

So with all of this down, why did I say it was “mostly successful”? Well, as it turns out, the tests work perfectly well when limited to problems involving planar motion. The results match up to those obtained by hand. But the results from the class get extremely nasty when dealing with more complex systems; I have implemented the rolling disc in two separate cases. In one test, I use the minimal set of generalized coordinates and the correct eoms are generated. But in another case I tried to use the non-minimal set of GCs and the equations generated are near impossible to comprehend (or I haven’t found the best way to deal with them yet). A big contribution of this messiness is due to the way in which Lagranges approach requires the definition of generalized speeds. In his approach, it is erquired for the generlized speeds to be ‘simple’ i.e. the gen. speeds are just derivatives of the gen coords. This is different in Kane’s approach where the generalized speeds can but needn’t necessarily be ‘simple’. From my experience, Kane’s generalized speeds are defined in a manner which make physical sense. This definitely validates why most dynamicists today (or so I have heard) prefer to choose Kane’s method on complex multi-body systems. The only way I can think of circumventing this situation in teh ‘LagrangesMethod’ class right now is using the minimal set of GCs for well known systems like the rolling disc and hope for the best.

Having all the additional functionality in this class and also playing with the rolling disc in particular has definitely led to a lot of insight but also taken a good chunk of time away from a period I wanted to dedicate to the ‘code output section’ which I have been unable to get started on. It looks like I will be unable to meet that one goal by the ‘hard’ pencils down date as I complete and fine tune the documentation (pending final approval of the Lagrange PR, of course). But I do feel that the time spent on Lagrange has been for the good. The code, I personally feel, is easy to read and appears to be easy to use. With people’s comments I was able to weed out all the unnecessary stuff. It is also ‘complete’ like I previously highlighted. I will continue to work on ‘code output’ post the GSoC period though as it’s usefulness is undeniable and also because of a development of a general sense of interest in coding (surprise surprise!).

Anyhow, apart from this, the other stuff I got done this week- I wrote up minor functions to compute a Lagrangian, changed how the potential energy function behaves. That’s it for this week. See you next week, one last time.