Riemannian cubic polynomials are a generalization of splines to Riemannian manifolds with applications in optimal control problems and interpolation problems. Riemannian cubic polynomials are obtained as the curves minimizing a higher-order functional depending on the covariant acceleration of the curve. Necessary conditions are complicated fourth order differential equations depending on geometric objects such as the Riemannian curvature tensor.
The numerical integration of this type of trajectories is a complex task because of the complexity of the geometric objects involved and the difficulty to maintain numerical solutions inside the desired manifold.
In this talk, we will expose our first steps towards defining a geometric integrator for Riemannian cubic polynomials. This research line leads to viewing Riemannian cubic polynomials as trajectories of higher-order Euler-Lagrange equations.
An essential ingredient of our method is the use of Retractions maps. They are used to consistently define a discretization of the tangent bundle as two copies of the configuration manifold. Such discretization maps can be conveniently lifted to a higher-order tangent bundle to construct geometric integrators for the higher-order Euler-Lagrange equations.
Along the talk, we will introduce both higher-order Euler-Lagrange equations as well as the notion of discretization maps associated with a retraction map to construct geometric integrators for mechanical systems. If time permits, we will use these techniques to show how to construct a geometric integrator for the Riemannian cubic polynomials in a sphere.