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How to find separation coordinates for the Hamilton–Jacobi equation: *a criterion of separability for natural hamiltonian systems*PrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2003 (English)In: Mathematical physics, analysis and geometry, ISSN 1385-0172, E-ISSN 1572-9656, Vol. 6, no 4, p. 301-348Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

2003. Vol. 6, no 4, p. 301-348
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:liu:diva-23159DOI: 10.1023/B:MPAG.0000007238.37788.2cLocal ID: 2563OAI: oai:DiVA.org:liu-23159DiVA, id: diva2:243473
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt446",{id:"formSmash:j_idt446",widgetVar:"widget_formSmash_j_idt446",multiple:true}); Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2017-12-13
##### In thesis

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.

1. Determination of separation coordinates for potential and quasi-potential Newton systems$(function(){PrimeFaces.cw("OverlayPanel","overlay243482",{id:"formSmash:j_idt720:0:j_idt724",widgetVar:"overlay243482",target:"formSmash:j_idt720:0:parentLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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