Some Results On The P-elastica

This paper deals with the regular curves in a Riemannian manifold with constant sectional curvature and the affine starlike curves in R2, R3 and R4. The p-elastica, the critical point for the total polynomial curvature functional on those curves with a fixed length satisfying given boundary conditions, is discussed. For the regular curves, we find two Killing fields for the purpose of integrating the structural equations of the p-elastic curves and express the p-elastica by quadratures in a system of cylindrical coordinates. For the star-like affine curves, we solve the Euler-Lagrange equation by quadratures and reduced the higher order structure equation to a first order linear system by using Killing field and the classification of linear Lie algebra sl(2, R) , sl(3,R) and sl(4, R). We solve the centroaffine p-elastica completely by quadratures.