We are developing homogeneous variational integrators on the two-sphere, which are based on Lie group variational integrators that constrain the algebra elements to the complementary space to the isotropy subspace of the group action of SO(3) on S^2. These yield exceptionally efficient global algorithms for Hamiltonian flows on S^2.
Stay tuned for the preprint on this, Lagrangian Mechanics and Variational Integrators on Two-Spheres, with Taeyoung Lee and N. Harris McClamroch.
In the meanwhile, please enjoy the following animations of some preliminary simulations we have done:
3-body problem on the sphere
Double spherical pendulum
Geometrically exact elastic rod model
Spherical pendula networks coupled with linear springs
2-dimensional array of magnets
Preprint is now available:
[ PDF | arXiv:0707.0022 [math.NA] ]
Invited article for the Springer Encyclopedia of Complexity and System Science
arxiv:0705.3868 [math.OC]
Discrete control systems, as considered here, refer to the control theory of discrete-time Lagrangian or Hamiltonian systems. These discrete-time models are based on a discrete variational principle, and are part of the broader field of geometric integration. Geometric integrators are numerical integration methods that preserve geometric properties of continuous systems, such as conservation of the symplectic form, momentum, and energy. They also guarantee that the discrete flow remains on the manifold on which the continuous system evolves, an important property in the case of rigid-body dynamics.
In nonlinear control, one typically relies on differential geometric and dynamical systems techniques to prove properties such as stability, controllability, and optimality. More generally, the geometric structure of such systems plays a critical role in the nonlinear analysis of the corresponding control problems. Despite the critical role of geometry and mechanics in the analysis of nonlinear control systems, nonlinear control algorithms have typically been implemented using numerical schemes that ignore the underlying geometry.
The field of discrete control systems aims to address this deficiency by restricting the approximation to choice of a discrete-time model, and developing an associated control theory that does not introduce any additional approximation. In particular, this involves the construction of a control theory for discrete-time models based on geometric integrators that yields numerical implementations of nonlinear and geometric control algorithms that preserve the crucial underlying geometric structure.