The Euler-Lagrange method for biharmonic curves
In this paper the authors consider biharmonic curves on a Riemannian manifold \((M,g)\), that is curves that solve the differential equation \[ \nabla^3_{\gamma^{\prime}}\gamma^{\prime}- R(\gamma^{\prime},\nabla_{\gamma^{\prime}}\gamma^{\prime})\gamma^{\prime}=0, \] where \(R\) denotes the curvature tensor of the Riemannian manifold \((M,g)\). Many papers have been recently devoted to biharmonic curves on surfaces and three--dimensional Riemannian manifolds. In this paper the authors prove a variational principle, showing that biharmonic curves are the critical points of a \textit{bienergy functional} \[ E_2(\gamma) = \int_a^b L_2(\gamma,\gamma^{\prime},\gamma^{\prime\prime})\,dt \] where \(L_2\) is a \textit{bilagrangian function} that depends on \(\gamma\), \(\gamma^{\prime}\), \(\gamma^{\prime\prime}\). So, the biharmonic equation represents the Euler--Lagrange equation for the functional \(E_2\). As application of this variational principle, the authors show that biharmonic curves parametrized by arc--length in the three--dimensional Lie group Sol are geodesics. Moreover parameterizations of biharmonic curves in the Cartan-Vranceanu three--dimensional spaces are exhibited, in particular in the three-dimensional Heisenberg group \(H_3\) endowed with a left invariant metric.
scientific article; zbMATH DE number 5147138