Variational principles on principal fiber bundles: A geometric theory of Clebsch potentials and Lin constraints (Q1107187)

The authors extend, from a product of a vector space and a Lie group to a nontrivial principal bundle P with structure group G, the variational principle involving Clebsch potentials given by the first and the third author [Physica D 27, 63-89 (1987; Zbl 0625.58037)]. The generalization is achieved by introducing the horizontal Lin constraints with respect to a connection on P and using then a Lagrange multiplier argument. The obtained variational principle is used to find equations of motion for a Lagrangian on P which is invariant under the tangent lifting of the action of G on P. As an application, it is studied the motion of a particle in a Yang-Mills field, where the Lagrangian is the kinetic energy.