Deformation of piecewise differentiable curves in constrained variational calculus

A survey of the geometric tools involved in the study of constrained variational calculus is presented. The central issue is the characterization of the admissible deformations of piecewise differentiable sections of a fibre bundle Vn+1→R, in the presence of arbitrary non-holonomic constraints. Asynchronous displacements of the corners are explicitly considered. The coordinate-independent representation of the variational equation and the associated concepts of infinitesimal control and absolute time derivative are reviewed. In the resulting algebraic environment, every admissible section is assigned a corresponding abnormality index, identified with the co-rank of a suitable linear map. Sections with vanishing index are called normal. A section is called ordinary if every solution of the variational equation vanishing at the endpoints is tangent to some finite deformation with fixed endpoints. The interplay between abnormality index and ordinariness - in particular the fact that every normal evolution is automatically an ordinary one - is discussed.