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- Rauch, Stefan, 1950-, et al.
(author)
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Stäckel separability for Newton systems of cofactor type
- 2004
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Other publication (other academic/artistic)abstract
- 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)-1∇W(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.
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| 2. |
- Rauch, Stefan, 1950-, et al.
(author)
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What an effective criterion of separability says about the Calogero type systems
- 2005
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In: Journal of Nonlinear Mathematical Physics. - : Springer Science and Business Media LLC. - 1402-9251 .- 1776-0852. ; 12:SUPPL. 1, s. 535-547
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Journal article (peer-reviewed)abstract
- In [15] we have proved a 1-1 correspondence between all separable coordinates on Rn (according to Kalnins and Miller [9]) and systems of linear PDEs for separable potetials V (q). These PDEs, after introducing parameters reflecting the freedom of choice of Euclidean reference frame, serve as an effective criterion of separability. This means that any V (q) satisfying these PDEs is separable and that the separation coordinates can be determined explicitly. We apply this criterion to Calogero systems of particles interacting with each other along a line.
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| 3. |
- Waksjö, Claes, 1971-
(author)
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Determination of separation coordinates for potential and quasi-potential Newton systems
- 2003
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Doctoral thesis (other academic/artistic)abstract
- 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)-1∇W(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.
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| 4. |
- Waksjö, Claes, 1971-, et al.
(author)
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How to find separation coordinates for the Hamilton–Jacobi equation : a criterion of separability for natural hamiltonian systems
- 2003
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In: Mathematical physics, analysis and geometry. - 1385-0172 .- 1572-9656. ; 6:4, s. 301-348
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Journal article (peer-reviewed)abstract
- 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.
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| 6. |
- Waksjö, Claes, 1971-
(author)
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Stäckel multipliers in Euclidean space
- 2000
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Licentiate thesis (other academic/artistic)abstract
- In order to apply the method of separation of variables to the natural Hamilton-Jacobi equation in Euclidean space, one has to find new curvilinear coordinates in which the transformed equation admits a complete separated solution . For a potential V(q) given in Cartesian coordinates, the main difficulty is to decide if such a transformation x(q) exists and to effectively 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 results applies to the Helmholtz (stationary Schrödinger) equation as well.
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