Canonical transformations associated with second order problems in the calculus of variations

Summary Our object is a systematic investigation of some of the properties of canonical transformations associated with second order problems in the calculus of variations. After the introduction of such transformations, together with the related concepts of Lagrange and Poisson brackets, the bracket relationships are found which characterize canonical transformations. This characterization is also achieved by means of so-called reciprocity relations between the original transformation and its inverse (which always exists). The effect of the canonical transformation on the underlying variational problem is discussed. It is also shown that the Jacobian of such a transformation always has the value unity. The special case when the canonical transformation is independent of the parameter (a generalization of the so-called time-independent canonical transformation of mechanics) is treated in some detail. Finally it is indicated how the present theory can be extended to problems of higher order. Some of the results of this paper are contained in a doctoral thesis ([2]) which was presented to the University of South Africa. The writer wishes to express his gratitude to his supervisor, ProfessorH. Rund, for his interest, encouragement and advice concerning this work.