## Variational Integrators

In my early days as an undergraduate student, I had a lot of fun writing C code to integrate ODE’s numerically, so I could simulate physical systems in a computer (integrating the pendulum ODE is a classic). However, I was not very happy to realize that traditional numerical methods very easily give results with no physical meaning.

For example, think of a simple pendulum oscillating on a vertical plane. Assuming that there are no damping forces, there’s energy conservation. However, traditional numerical methods (such as Runge-Kutta methods) do not, in general, preserve energy. The numerical simulation of the pendulum may thus exhibit non-physical energy dissipation (or amplification).

[ image courtesy of Ari Stern ]

When we integrate an ODE numerically, we obtain a discrete trajectory which we hope is “close enough” to the exact flow of the differential equation. In some problems in Physics, the continuous flow lies on a manifold, and we would like to preserve this geometric property: it would be wonderful if the discrete trajectory lied on the same manifold as the continuous flow. While thinking about numerical methods that preserve certain invariants, I became interested in the field of geometric integration.

One particular class of geometric integrators I have been interested in over the past year is the class of variational integrators. Many physical laws are formulated as variational principles from which differential equations can be derived. Instead of discretizing the differential equations, why not discretize the variational principles instead? This rather sexy thought lies at the basis of the nascent field of variational integrators.

Here’s an overview on variational integrators, by Prof. Matthew West. I have been reading some papers on the topic. This field seems to have been explored rather vigorously at Caltech, as there are/were a lot of people working on it:

and their collaborators. I have compiled a list of lecture slides, theses and papers on the area of variational integrators. Hope you find the list useful (I sure do!).

-/-

Some lecture slides for a gentle introduction:

-/-

A few PhD theses on variational integrators and their application:

-/-

Some papers on the foundations of variational integrators:

Applications to Computational Mechanics:

Applications to Optimal Control Theory:

Applications to Computational Electromagnetism:

Applications to Computational Physics:

The papers I have listed so far refer to deterministic integrators. However, quite recently Prof. Houman Owhadi and Nawaf Bou-Rabee have been working on stochastic variational integrators:

I wonder if stochastic variational integrators could be used in quantitative finance. Just a wild thought…

-/-

If you know some more papers, please let me know! Last but not least, Ganesh Swami once wrote a brief post on variational integrators.