Chapter 2 Discrete Routh Reduction

This chapter investigates the relationship between Routh symmetry reduction and time discretization for Lagrangian systems. Within the framework of discrete variational mechanics, a discrete Routh reduction theory is constructed for the case of abelian group actions, and extended to systems with constraints and non-conservative forcing or dissipation. Variational Runge–Kutta discretizations are considered in detail, including the extent to which symmetry reduction and discretization commute. In addition, we obtain the Reduced Symplectic Runge–Kutta algorithm, which can be considered a discrete analogue of cotangent bundle reduction. We demonstrate these techniques numerically for satellite dynamics about the Earth with a non-spherical J2 correction, and the double spherical pendulum. The J2 problem is interesting because in the unreduced picture, geometric phases inherent in the model and those due to numerical discretization can be hard to distinguish, but this issue does not appear in the reduced algorithm, and one can directly observe interesting dynamical structures. The main point of the double spherical pendulum is to provide an example with a nontrivial magnetic term in which our method is still efficient, but is challenging to implement using a standard method. 2.