Related papers: Symbolic software for separation of variables in t…
We review the theory of orthogonal separation of variables of the Hamilton-Jacobi equation on spaces of constant curvature, highlighting key contributions to the theory by Benenti. This theory revolves around a special type of conformal…
We address the problem of the separation of variables for the Hamilton-Jacobi equation within the theoretical scheme of bi-Hamiltonian geometry. We use the properties of a special class of bi-Hamiltonian manifolds, called omega-N manifolds,…
We show that with every separable calssical Stackel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing…
Geometric separability theory of Gel'fand-Zakharevich bi-Hamiltonian systems on Riemannian manifolds is reviewed and developed. Particular attention is paid to the separability of systems generated by the so-called special conformal Killing…
The Separation of Variables theory for the Hamilton-Jacobi equation is 'by definition' related to the use of special kinds of coordinates, for example Jacobi coordinates on the ellipsoid or St\"ackel systems in the Euclidean space. However,…
We study concircular tensors in spaces of constant curvature and then apply the results obtained to the problem of the orthogonal separation of the Hamilton-Jacobi equation on these spaces. Any coordinates which separate the geodesic…
We discuss from a bi-Hamiltonian point of view the Hamilton-Jacobi separability of a few dynamical systems. They are shown to admit, in their natural phase space, a quasi-bi-Hamiltonian formulation of Pfaffian type. This property allows us…
Liouville integrable separable systems with quadratic in momenta first integrals are considered. Particular attention is paid to the systems generated by the so-called special conformal Killing tensors, i.e. Benenti systems. Then,…
We review the theory of orthogonal separation of variables on pseudo-Riemannian manifolds of constant non-zero curvature via concircular tensors and warped products. We then apply this theory simultaneously to both the three-dimensional…
Separable coordinate systems are introduced in the complex and real four-dimensional flat spaces. We use maximal Abelian subgroups to generate coordinate systems with a maximal number of ignorable variables. The results are presented (also…
This is a continuation of the work initiated in a previous paper on so-called driven cofactor systems, which are partially decoupling second-order differential equations of a special kind. The main purpose in that paper was to obtain an…
For a class of Hamiltonian systems naturally arising in the modern theory of separation of variables, we establish their maximal superintegrability by explicitly constructing the additional integrals of motion.
We look for spectral type differential equations for the generalized Jacobi polynomials and for the Sobolev-Laguerre polynomials. We use a method involving computeralgebra packages like Maple and Mathematica and we will give some…
This survey examines separation of variables for algebraically integrable Hamiltonian systems whose tori are Jacobians of Riemann surfaces. For these cases there is a natural class of systems which admit separations in a nice geometric…
We discuss the Boussinesq system with $t_5$ stationary, within a general framework for the analysis of stationary flows of n-Gel'fand-Dickey hierarchies. We show how a careful use of its bihamiltonian structure can be used to provide a set…
We discuss a general procedure for arriving at the Hamilton-Jacobi equation of second-class constrained systems, and illustrate it in terms of a number of examples by explicitely obtaining the respective Hamilton principal function, and…
We study Killing tensors in the context of warped products and apply the results to the problem of orthogonal separation of the Hamilton-Jacobi equation. This work is motivated primarily by the case of spaces of constant curvature where…
The Neumann system on the 2-dimensional sphere is used as a tool to convey some ideas on the bi-Hamiltonian point of view on separation of variables. It is shown that, from this standpoint, its separation coordinates and its integrals of…
An outline of the basic Riemannian structures underlying the separation of variables in the Hamilton-Jacobi equation of natural Hamiltonian systems.
Recently the Hamilton-Jacobi formulation for first order constrained systems has been developed. In such formalism the equations of motion are written as total differential equations in many variables. We generalize the Hamilton-Jacobi…