Ueber die Integration der Hamilton'schen Differentialgleichung mittelst Separation der Variabeln

Stäckel, Paul

doi:10.1007/bf01445366

Cited by 69 publications

(61 citation statements)

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“…Theorem 1.5 ( [13,45]). On a Riemannian manifold, there is a bijective correspondence between orthogonal separation coordinates and Stäckel systems.…”

Section: The Problemmentioning

confidence: 99%

The Variety of Integrable Killing Tensors on the 3-Sphere

Schöbel¹

2014

SIGMA

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Abstract. Integrable Killing tensors are used to classify orthogonal coordinates in which the classical Hamilton-Jacobi equation can be solved by a separation of variables. We completely solve the Nijenhuis integrability conditions for Killing tensors on the sphere S 3 and give a set of isometry invariants for the integrability of a Killing tensor. We describe explicitly the space of solutions as well as its quotient under isometries as projective varieties and interpret their algebro-geometric properties in terms of Killing tensors. Furthermore, we identify all Stäckel systems in these varieties. This allows us to recover the known list of separation coordinates on S 3 in a simple and purely algebraic way. In particular, we prove that their moduli space is homeomorphic to the associahedron K 4 .

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“…Theorem 1.5 ( [13,45]). On a Riemannian manifold, there is a bijective correspondence between orthogonal separation coordinates and Stäckel systems.…”

Section: The Problemmentioning

confidence: 99%

The Variety of Integrable Killing Tensors on the 3-Sphere

Schöbel¹

2014

SIGMA

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“…The motivation starts with the following observation: The classical approach to finding the relevant integrals was based on solving the Hamilton-Jacobi equation by separation of variables (Stackel [19], Jacobi [6]). This required the appropriate choice of variables and computational skill.…”

Section: Introduction a Backgroundmentioning

confidence: 99%

Geometry of Quadrics and Spectral Theory

Moser

1980

The Chern Symposium 1979

199

234

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“…Stäckel [14] formulated a necessary and sufficient condition for separability of natural Hamiltonians. Levi-Civita [10] found a system of equations to be satisfied by any separable Hamiltonian.…”

Section: The Problemmentioning

confidence: 99%

“…The classical algebraic condition for an orthogonal coordinate system to be separable is that the metric is in Stäckel form [14]. An equivalent condition is that the metric satisfies a certain system of PDEs given by Levi-Civita [10].…”

Section: Separable Coordinate Systemsmentioning

confidence: 99%

What an Effective Criterion of Separability says about the Calogero Type Systems

Rauch-Wojciechowski¹,

Waksjö²

2005

JNMP

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In [15] we have proved a 1-1 correspondence between all separable coordinates on R n (according to Kalnins and Miller [9]) and systems of linear PDEs for separable potentials 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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Ueber die Integration der Hamilton'schen Differentialgleichung mittelst Separation der Variabeln

Abstract: In der &bhandlung: Eine charaktvristische ~,igenschaft der El~he~, deren I.,inienelement d s durch d8~ = (~(q,) + l(q~)) (~ql ~ + ~q~) gegebe~ wird (diese Annalen t Bd. 35, 1889, S. 91--103) habe ich die Frage, warm eine ttami~on'sc]~e ~DifferegtiaO~:

Cited by 69 publications

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The Variety of Integrable Killing Tensors on the 3-Sphere

The Variety of Integrable Killing Tensors on the 3-Sphere

Geometry of Quadrics and Spectral Theory

What an Effective Criterion of Separability says about the Calogero Type Systems

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