Lagrangians and sand

2009 November 22

So you have a “sand statistics manifold” and you’d like to figure out what happens when you shake it with an earthquake. What do you do? That’s the real question. Part one is that we don’t know which statistics we need, and part two is we don’t know which equations will describe their changes in time.

To answer those questions, I’ve been turning to Lagrangian Mechanics. Basically, Lagrangian mechanics is involved with describing how a classical physical system behaves in terms of some finite set of parameters called the state, and a function of those parameters called the Lagrangian. It is a generalization of Newton’s laws. For Classical Mechanics, the Lagrangian is basically the total kinetic energy minus the total potential energy. More on that either at Wikipedia, or the excellent book by Sussman and Wisdom.

If we can calculate the total kinetic and potential energy of the system in terms of the statistics we’ve got spread around through space, then we can get a relationship describing the dynamics of the system. This differs significantly from the classical “continuum theory” of materials behavior in that it is more explicit about how the “continuum” arises. Continua are models, they are not reality since everything is composed of a finite number of atoms.

So what I need now is a way to describe the amount of kinetic and potential energy in a continuum that models sand, and a way to describe that continuum in terms of a finite set of numbers (the coordinates), and then probably some constraints on how the continuum can move so that it best approximates the real sand variables.

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