A simple derivation and interpretation of the third integral in stellar dynamics

Lynden-Bell, D.

doi:10.1046/j.1365-8711.2003.06041.x

Cited by 11 publications

(22 citation statements)

References 12 publications

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“…This integral of motion -initially we shall call it 'Euler's third integral'-is derived by applying the Hamilton-Jacobi method when we perform separation of variables in a suitable coordinate system [16]. Although, this 3rd integral of motion is known for more than a century, quite recently, Lynden-Bell [17], trying to explain its physical meaning, offered a simple and straightforward constructive method to build it. He proved that its kinetic part is the scalar product of the angular momenta about the two centers of mass and defined it as:…”

Section: The Euler Problem 21 the Original Problemmentioning

confidence: 99%

Newtonian analogue of a Kerr black hole

Areti

Apostolatos

2020

Phys. Rev. D

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A 250-year old Newtonian problem, first studied by Euler, turns out to share a lot of similarities with the most extreme astrophysical relativistic object, the Kerr black hole. Although the framework behind the two fields is completely different, both problems are related to gravitational fields that have quite intriguing analogies with respect to orbital motions of a test-body in them. The fundamental reason responsible for their extraordinary similarity is the integrability of both problems, as well as their common multipolar structure. In this paper we demonstrate the existence of a multitude of either qualitative, and sometimes quantitative, similarities between the two problems. Based on this analogy, one could use the Newtonian problem to get insight in cases where the relativistic treatment of the field of a Kerr black hole becomes quite complicated. *

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Section: The Euler Problem 21 the Original Problemmentioning

confidence: 99%

Newtonian analogue of a Kerr black hole

Areti

Apostolatos

2020

Phys. Rev. D

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“…(36). The potential (33) can be considered the Newtonian analogue (Keres 1967;Israel 1970;Lynden-Bell 2003;) of the Kerr solution in general relativity. The analogy is manifest when the Kerr metric is written in Kerr-Schild coordinates (Kerr & Schild 1965).…”

Section: Existence and Uniqueness Of The Quadratic Invariantmentioning

confidence: 99%

“…The direct approach employed here is more algorithmic and computationally intensive compared to other approaches (Lynden-Bell 2003;). But the advantages of the direct approach are that it establishes uniqueness and that it can be straightforwardly generalized to higher order or relativistic invariants.…”

Section: Cosmological Constant: Analogy With Amentioning

confidence: 99%

Constants of motion in stationary axisymmetric gravitational fields

Markakis

2014

Monthly Notices of the Royal Astronomical Society

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The motion of test particles in stationary axisymmetric gravitational fields is generally nonintegrable unless a nontrivial constant of motion, in addition to energy and angular momentum along the symmetry axis, exists. The Carter constant in Kerr-de Sitter spacetime is the only example known to date. Proposed astrophysical tests of the black-hole no-hair theorem have often involved integrable gravitational fields more general than the Kerr family, but the existence of such fields has been a matter of debate. To elucidate this problem, we treat its Newtonian analogue by systematically searching for nontrivial constants of motion polynomial in the momenta and obtain two theorems.First, solving a set of quadratic integrability conditions, we establish the existence and uniqueness of the family of stationary axisymmetric potentials admitting a quadratic constant. As in Kerr-de Sitter spacetime, the mass moments of this class satisfy a "no-hair" recursion relation M 2l+2 = a 2 M 2l , and the constant is Noetherrelated to a second-order Killing-Stäckel tensor. Second, solving a new set of quartic integrability conditions, we establish nonexistence of quartic constants. Remarkably, a subset of these conditions is satisfied when the mass moments obey a generalized "no-hair" recursion relation M 2l+4 = (a 2 + b 2 )M 2l+2 − a 2 b 2 M 2l . The full set of quartic integrability conditions, however, cannot be satisfied nontrivially by any stationary axisymmetric vacuum potential.

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“…h i = r i × v with v the particle's velocity. It should be noted that the explicit form of Q can be derived in more than one ways: for example, Lynden-Bell [7] provides a straightforward derivation of this result, while Ref. [11] arrives at the same result by separating the Hamilton-Jacobi equation.…”

Section: The Analogue Of Carter's Constant In Newtonian Gravitymentioning

confidence: 99%

The separable analogue of Kerr in Newtonian gravity

Glampedakis

Apostolatos

2013

Class. Quantum Grav.

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General Relativity's Kerr metric is famous for its many symmetries which are responsible for the separability of the Hamilton-Jacobi equation governing the geodesic motion and of the Teukolsky equation for wave dynamics. We show that there is a unique stationary and axisymmetric Newtonian gravitational potential that has exactly the same dual property of separable point-particle and wave motion equations. This 'Kerr metric analogue' of Newtonian gravity is none other than Euler's 18th century problem of two-fixed gravitating centers.

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A simple derivation and interpretation of the third integral in stellar dynamics

Abstract: Starting from the problem of two fixed centres we find a simple derivation of its third integral in terms of the scalar product of the angular momenta about the two fixed centres. This is then generalized to find the general form of the potential in which an exact third integral exists.

Cited by 11 publications

References 12 publications

Newtonian analogue of a Kerr black hole

Newtonian analogue of a Kerr black hole

Constants of motion in stationary axisymmetric gravitational fields

The separable analogue of Kerr in Newtonian gravity

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