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Determination of separation coordinates for potential and quasi-potential Newton systems
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
2003 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

When solving Newton systems q = M(q), q ϵ Rn, by the method of separation of variables, one has to determine coordinates in which the related Hamilton-Jacobi equation separates.

The problem of finding separation coordinates for potential Newton systems q = -V (q) goes back ta Jacobi. In the first part of this thesis we give a complete solution to this classical problem. It can also be used to find separation coordinates for the Schrödinger equation.

In the second part of this thesis, we study separability for quasi-potential systems q = -A(q)-1W(q) of generic cofactor pair type. We define separation coordinates that give these systems separable Stäckel form. The two most important families of these coordinates (cofactor-elliptic and cofactor-parabolic) generalize the Jacobi elliptic coordinates, and are shown to be defines by elegant rational equations.

Place, publisher, year, edition, pages
Linköping: Linköpings universitet , 2003. , 18 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 845
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-23168Local ID: 2573ISBN: 91-7373-755-0 (print)OAI: oai:DiVA.org:liu-23168DiVA: diva2:243482
Public defence
2003-11-07, Sal C3, Hus C, Linköpings Universitet, Linköping, 10:15 (Swedish)
Opponent
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-01-07
List of papers
1. How to find separation coordinates for the Hamilton–Jacobi equation: a criterion of separability for natural hamiltonian systems
Open this publication in new window or tab >>How to find separation coordinates for the Hamilton–Jacobi equation: a criterion of separability for natural hamiltonian systems
2003 (English)In: Mathematical physics, analysis and geometry, ISSN 1385-0172, E-ISSN 1572-9656, Vol. 6, no 4, 301-348 p.Article in journal (Refereed) Published
Abstract [en]

The method of separation of variables applied to the natural Hamilton–Jacobi equation ½ ∑(∂u/∂q i )2+V(q)=E consists of finding new curvilinear coordinates x i (q) in which the transformed equation admits a complete separated solution u(x)=∑u (i)(x i ;α). For a potential V(q) given in Cartesian coordinates, the main difficulty is to decide if such a transformation x(q) exists and to determine it explicitly. Surprisingly, this nonlinear problem has a complete algorithmic solution, which we present here. It is based on recursive use of the Bertrand–Darboux equations, which are linear second order partial differential equations with undetermined coefficients. The result applies to the Helmholtz (stationary Schrödinger) equation as well.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-23159 (URN)10.1023/B:MPAG.0000007238.37788.2c (DOI)2563 (Local ID)2563 (Archive number)2563 (OAI)
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-01-07
2. Stäckel separability for Newton systems of cofactor type
Open this publication in new window or tab >>Stäckel separability for Newton systems of cofactor type
2004 (Swedish)Manuscript (preprint) (Other academic)
Abstract [en]

A conservative Newton system ¨q = -∇V(q) in Rnis called separable when the Hamilton-Jacobi equation for the Natural Hamiltonian H = ½p2+ V (q) can be solved through separation of variables in some curvilinear coordinates. If these coordinates are orthogonal, the Newton system admits n first integrals, which all have separable Stäckel form with quadratic dependence on p.

We study here separability of the more general class of Newton systems ¨q = - (cof G)-1W(q) that admit n quadratic first integrals. We prove that a related system with the same integrals can be transformed through a non-canonical transformation into a Stäckel separable Hamiltonian system and solved by quadratures, providing a solution to the original system.

The separation coordinates, which are defined as characteristic roots of a linear pencil G - μ~G of elliptic coordinates matrices, generalize the well known elliptic and parabolic coordinates. Examples of such new coordinates in two and three dimensions are given.

These results extend, in a new direction, the classical separability theory for natural Hamiltonians developed in the works if Jacobi, Liouville, Stäckel, Levi-Civita, Eisenhart, Bebenti, Kalnins and Miller.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-22694 (URN)1989 (Local ID)1989 (Archive number)1989 (OAI)
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-01-07

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Waksjö, Claes

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