domingo, 14 de marzo de 2010

LAGRANGIAN DYNAMICS

These days I've been researching on Lagrangian Dynamics.

Finally, some light has been shed through the thick mesh of formulae, technical and mathematical verbiage, and an atom of understanding has entered into my banged head.

The paper that has finally put everything together is this:

http://box2d.googlecode.com/files/GDC2009_ErinCatto.zip

But before, I have had to go through an odissey of other papers to understand why they call it Lagrangian Dynamics, and Lagrangian by extension, to everything related to dynamics in the latest physics simulation R+D world.

The thing is as follows, explained in plane English for plane human beings (like myself, ahem...):

Classical Newtonian mechanics have some limitations when it comes to computing the equations of motion.
We want to iteratively solve a set of Ordinary Differential Equations (ODE) for acceleration, velocity and position if we want to know where our physical elements will be.
But it happened that mysteriously, when computing them, the total energy of the system got affected, because dissipative forces like damping, friction, etc... had to be also implemented, at that is a daunting task...

And there is where Lagrange comes into play: he theorized that we can make use of a further abstraction of our system and embed it into another, a bit broader conceptually, where all these forces would not be so determinant and we still could retrieve our results for position, by applying energetic conservation concepts.
Of course this abstraction comes at the price of having to understand a lot of many other things, not simple either (one funny thing about all this is that the only examples I have been able to find are that of a pendulum, a spring tied to a mass and a bead on a wire...If you didn't get it at the beginning, you'll never do...until you get it. Then these are so obvious you can't think of another!)

Anyway, the main point on Lagrangian approach to dynamics is that we can (and we normally do) find a set of equations that constraint the movement so the energy of the particles individually remains untouched.
These constraints materialize in the form of Jacobian Matrices or Lagrange Multipliers applied to each particle, rigid body or mass in a mass-spring system into our simulation (this is much deeper explained in the paper).

Fortunatelly also, all the above can be applied to Finite Element Method meshes. All we have to consider is that, in Lagrangian dynamical FEM, instead of having only one term for the stiffness matrix as usual:

                     [F]=[K]·[u]

a few more matrices that represent the dynamic terms appear:

                [M][u''] + [C][u'] + [K][u] = [F]



And this is it regarding brand new concepts. It is as far as I can go by now.

In the way I have encountered many intensely related diverse topics, which should be explained in following posts. Some of them are:
  • Linear Complementary Problem Solvers
  • Principle of Virtual Work
  • Differential Equations
  • Smoothed Particle Hydrodynamics (SPH)
  • Kynematics
But for now I will leave this introduction as it is

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