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Stäckel separability for Newton systems of cofactor type
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
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.

Place, publisher, year, edition, pages
2004.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-22694Local ID: 1989OAI: oai:DiVA.org:liu-22694DiVA: diva2:243007
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-01-07
In thesis
1. Determination of separation coordinates for potential and quasi-potential Newton systems
Open this publication in new window or tab >>Determination of separation coordinates for potential and quasi-potential Newton systems
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:nbn:se:liu:diva-23168 (URN)2573 (Local ID)91-7373-755-0 (ISBN)2573 (Archive number)2573 (OAI)
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

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Rauch, StefanWaksjö, Claes

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Citation style
  • apa
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