Ganesh Swami

Dynamical systems following Hamilton mechanics can be formulated as

where M is the mass and q is the state vector. These equations arise in pretty much every single physics engine (astrophysics and molecular dynamics for example.) Traditionally, solvers such as the forward/backward Euler have been used, but these solvers do not respect the manifold of the configuration system. This led to the development of solvers such as the Verlet (and Velocity Verlet) which are symplectic in nature and follow the geometry of the problem. The Velocity Verlet specifically is the workhorse of molecular dynamics.

By geometry, I mean the invariants that arise due to symmetry in the system. For example, rotational and translational symmetry in the system give rise to conservation of angular and linear momentum. Above, the equations of motion for a simple pendulum were integrated with four different solvers. The symplectic nature of the problem does not fall out naturally from the solvers because these “local” methods still look at differential changes in momenta and position.

Variational methods, on the other hand, directly deal with equations arising out of Hamilton’s action principle. The Lagrangian is defined as

and the action functional is the integral of the Lagrangian along a path q(t).

First order variations to compute the stationary action leads to the Euler-Lagrange differential equation. A similar derivation can be done for discrete variables yielding the Discrete Euler Lagrange (DEL) equation.

This is very attractive because physical invariants that arise from the variational principle are guaranteed to be maintained in the discrete situation as well. This is also true for constraints on the system: constraint versions of Verlet and Velocity Verlet, SHAKE and RATTLE are natural in the DEL equation setting.

The only roadblock to the rapid adoption of these algorithms is the computational expense of solving the DEL. Each integration timestep requires the solution of a set of implicit nonlinear equations (usually by the use of Newton’s method.) This is a problem for molecular dynamics where you are trying to run the simulation for a few nanoseconds and your timestep is in femtoseconds.

I was almost going to choose this topic for my nonlinear physics course project. Though this is very interesting and exciting, I couldn’t find enough content to develop into a project.

Granular media, like sand, rice or cereal, also happen to form interesting patterns when vibrated on a membrane. This was first discovered and published in Nature by Umbanhowar, Melo & Swinney from the Center for Nonlinear Dynamics at the University of Texas at Austin. This was the same team that had created the very popular Faraday waves video posted on YouTube.

The physical phenomena is similar to Faraday waves seen in liquids almost a hundred and thirty years ago. The key difference is that while liquid faraday waves can be derived from a continnum model such as the Navier-Stokes equations, no such model exists for granular media as far as my limited knowledge goes. Therefore, the analytical derivations for the purposes of a project is limited.

The phase-plot is in terms of a dimensionless amplitude given by

where A is the driving amplitude, g is the acceleration due to gravity, and f is the driving frequency. Some of these have been experimentally verified by means of a molecular dynamics simulation. Once again, the computational part wasn’t sufficient to fulfill the requirements because the equations are “simple” non-linear coupled ordinary differential equations. Whereas systems we have studied in class were described by partial differential equations.

The term “oscillon” was coined by the paper mentioned above. In recent years, there has been a lot of research into these structures. The picture above is oscillons in a nonlinear field model and the author has some really cool videos on his site.

I spent most of last week looking for an idea for my nonlinear physics course project. We were given a few pointers to topics that could become potential projects: various kinds of pattern formation in Fluid Mechanics, Complex fluids, Biological systems, Chemical systems, Optics, Combustion and Hydrodynamics.

I came up with four topics (all four of them weren’t on the initial list of things suggested.) Somebody else in the class had exactly the same proposal as mine, including references, so that was out of the picture. One turned out to be computationally infeasible (couple of hours for a single step of a Finite Element Method), the other turned out to be extremely hard and outside the scope of the course.

I finally settled on “Global feedback methods for the subcritical Complex Ginzburg-Landau Equation.” If that didn’t make any sense, I’ve tried to explain what each word means in my proposal.

(picture courtesy of Shankar Venkataramani.)