Variational and Optimal Control Approaches for the Second-Order Herglotz Problem on Spheres

Abstract

The present paper extends the classical second-order variational problem of Herglotz type to the more general context of the Euclidean sphere $$S^n$$Sn following variational and optimal control approaches. The relation between the Hamiltonian equations and the generalized Euler–Lagrange equations is established. This problem covers some classical variational problems posed on the Riemannian manifold $$S^n$$Sn such as the problem of finding cubic polynomials on $$S^n$$Sn. It also finds applicability on the dynamics of the simple pendulum in a resistive medium.