Invariants at fixed and arbitrary energy. A unified geometric approach

Abstract

Invariants at arbitrary and fixed energy (strongly and weakly conserved quantities) for 2-dimensional Hamiltonian systems are treated in a unified way. This is achieved by utilizing the Jacobi metric geometrization of the dynamics. Using Killing tensors we obtain an integrability condition for quadratic invariants which involves an arbitrary analytic function S(z). For invariants at arbitrary energy the function S(z) is a second degree polynomial with real second derivative. The integrability condition then reduces to Darboux's condition for quadratic invariants at arbitrary energy. The four types of classical quadratic invariants for positive definite 2-dimensional Hamiltonians are shown to correspond to certain conformal transformations. We derive the explicit relation between invariants in the physical and Jacobi time gauges. In this way knowledge about the invariant in the physical time gauge enables one to directly write down the components of the corresponding Killing tensor for the Jacobi metric. We also discuss the possibility of searching for linear and quadratic invariants at fixed energy and its connection to the problem of the third integral in galactic dynamics. In our approach linear and quadratic invariants at fixed energy can be found by solving a linear ordinary differential equation of the first or second degree respectively.

The Maupertuis Principle and Canonical Transformations of the Extended Phase Space

“…If t be the time along trajectories of the new vector field ξ, then the Maupertuis mapping gives rise to so-called Jacobi transformations [10,16,27] of the Hamilton function (2.2) and of the time variable…”

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].…”

Nonstandard Separability on the Minkowski Plane

“…They exist more general separable geodesic flows on a generic pseudo-Riemannian metric (2.2) and/or natural Hamiltonian systems separable for fixed values of H = E [10,12] in which the functions Σ = Σ(u + x) andΣ =Σ(u − x) are not restricted to the class (2.5). We observe that the procedure recalled in the present section applies to this general setting including fixed-energy ("weak") separability and is based on a generalization of the conformal coordinate transformation introduced by Kolokoltsov [13].…”

“…In particular, it is in general necessary to use different separating variables, even for the integration of a single orbit. Associating as usual the existence of a 2nd-rank Killing tensor to that of a system of separating coordinates [6], the picture can be illustrated as follows: for (1 + 1)-dimensional systems there are three possible types of conformal Killing tensors, and therefore, three distinct separability structures in contrast to the single standard (Liouville) type separation of the positive definite case [10]. One of the new separability structures is the complex-Liouville/harmonic type which is characterized by complex separation variables and the metric is an harmonic function.…”