The Maupertuis Principle and Canonical Transformations of the Extended Phase Space

Abstract

We discuss some special classes of canonical transformations of the extended phase space, which relate integrable systems with a common Lagrangian submanifold. Various parametric forms of trajectories are associated with different integrals of motion, Lax equations, separated variables and action-angles variables. In this review we will discuss namely these induced transformations instead of the various parametric form of the geometric objects.

Complex Representation of Planar Motions and Conserved Quantities of the Kepler and Hooke Problems

“…This relation has been rediscovered and generalized by Arnold and Vassiliev [1,2] in 1989 and quite simultaneously by Hojman et al [17]. In fact the generalization of Kasner's result had already been obtained by Collas [5] and implicitly enters into the frame of the coupling constant metamorphosis of Hietarinta et al [16,31,37]. Even if we restrict ourselves to the classical aspects (the study of this correspondence in the quantum mechanical frame has an interesting parallel history that we do not deal with here), numerous articles have been published on this subject during the last fifteen years [13,15,24,25,35,36,38].…”

Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta

“…Theorem 3 and Corollary 1 below bear some resemblance with other classes of transformations between dynamical systems [1,2,11,38,41,42,43]. However, the present result is of different nature and is deeper because, in order to construct the second system, one needs to know the quadratic integral of the first one.…”

Generalized Stäckel transform and reciprocal transformations for finite-dimensional integrable systems

“…In order to find a complete integral (36) we can use separation of variables as follows (see e.g. [33,7] and references therein).…”

“…[29,31] and references therein. On the other hand, some particular examples of transformations of this kind for finite-dimensional Hamiltonian systems are also known, for instance the Jacobi transformation, see [24] and a recent survey [36]. The reciprocal transformations of somewhat different kind have also appeared in [18,38,35].…”