An intrinsic version of the k-harmonic equation

Abstract

The notion of k-harmonic curves is associated with the kth-order variational problem defined by the k-energy functional. The present paper gives a geometric formulation of this higher-order variational problem on a Riemannian manifold M and describes a generalized Legendre transformation defined from the kth-order tangent bundle $T^kM$ to the cotangent bundle $T^*T^{k-1}M$. The intrinsic version of the Euler–Lagrange equation and the corresponding Hamiltonian equation obtained via the Legendre transformation are achieved. Geodesic and cubic polynomial interpolation is covered by this study, being explored here as harmonic and biharmonic curves. The relationship of the variational problem with the optimal control problem is also presented for the case of biharmonic curves.