Discrete Control Systems

Discrete variational mechanics: A formulation of mechanics in discrete-time that is based on a discrete analogue of Hamilton’s principle, which states that the system takes a trajectory for which the action integral is stationary. Geometric integrator: A numerical method for obtaining numerical solutions of differential equations that preserves geometric properties of the continuous flow, such as symplecticity, momentum preservation, and the structure of the configuration space. Lie group: A differentiable manifold with a group structure where the composition is differentiable. The corresponding Lie algebra is the tangent space to the Lie group based at the identity element. Symplectic: A map is said to be symplectic if given any initial volume in phase space, the sum of the signed projected volumes onto each position-momentum subspace is invariant under the map. One consequence of symplecticity is that the map is volume-preserving as well.