Beijing Institute of Mathematical Sciences and Applications

This course is a part of a bigger project related to what we call “geometrizing mechanics". The project includes a lot of stages all united under the same goal, to find relevant geometric formalism to describe the internal structure of differential equations governing physical systems, or to deduce those equations from the physical properties of a system via a geometric structure.

The most studied example of this approach involves conservative mechanical systems expressed in natural variables. A convenient formalism for this is the Hamiltonian mechanics, though it is very restrictive since we have to preserve symplectic/Poisson structure and Hamiltonian. This classical description dates back to Lagrange, Jacobi and Poincaré, although these works did not use the modern language of Souriau and Arnold.

When dissipation or interaction enters the play, another formalism is needed and we have now Dirac mechanics, Herglotz principle and vakonomic dynamics, Gauss-Hertz principle and nonholonomic mechanics, port-Hamiltonian mechanics, contact mechanics, optimal control theory, etc. This course aims to provide an overview of popular non-Hamiltonian equations of motion, their underlying geometric structures, and structure-preserving integrators.

We will start by recalling the classical Hamiltonian formalism, then we introduce some proper terminology for various non-Hamiltonian systems and describe the desired output of the procedure together with some methods and examples.