Stability of spherical stellar systems -- I. Analytical results

Perez, Jérôme; Aly, J. J.

doi:10.1093/mnras/280.3.689

Cited by 36 publications

(45 citation statements)

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“…In the case where the function F depends only on e = |v| 2 /2 + φ(x), a proof of this inequality can be found in [17,16,36,42]. In our context F depends on e = |v| 2 /2 + φ(x) and = |x × v| 2 , and for a sake of clarity and completeness, we give a proof of this inequality in Appendix B which is a simple extension of the proof in [16].…”

Section: Study Of the Reduced Functional Jmentioning

confidence: 95%

A new variational approach to the stability of gravitational systems

Lemou¹,

Méhats²,

Raphaël³

2009

Comptes Rendus. Mathématique

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Abstract. We consider the three dimensional gravitational Vlasov Poisson system which describes the mechanical state of a stellar system subject to its own gravity. A well-known conjecture in astrophysics is that the steady state solutions which are nonincreasing functions of their microscopic energy are nonlinearly stable by the ow. This was proved at the linear level by Antonov in 1961. Since then, standard variational techniques based on concentration compactness methods as introduced by P.-L. Lions in 1983 have led to the nonlinear stability of subclasses of stationary solutions of ground state type.In this paper, we propose a new variational approach based on the minimization of the Hamiltonian under equimeasurable constraints which are conserved by the nonlinear transport ow, and recognize any anisotropic steady state solution which is a decreasing function of its microscopic energy as a local minimizer. The outcome is the proof of its nonlinear stability of under radially symmetric perturbations.

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Section: Study Of the Reduced Functional Jmentioning

confidence: 95%

A new variational approach to the stability of gravitational systems

Lemou¹,

Méhats²,

Raphaël³

2009

Comptes Rendus. Mathématique

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“…Consider the first‐order variation of an equilibrium . It is well known (see Bartolomew 1971; Kandrup 1990; Perez & Aly 1996) that there exists a phase‐space function , such that This function is called a generator 2 of the perturbation. Written in this form, f (1) is the largest class of physical perturbations which can be considered as acting on f 0 .…”

Section: Dissipation‐induced Instabilities and Self‐gravitating Sysmentioning

confidence: 99%

“…Any stationary solution of the CBP system is associated to an equilibrium state of the particles distribution. It is now well known that CBP system is Hamiltonian with respect to a Poisson bracket of non‐canonical form arising from the fact that a distribution function does not constitute a set of canonical field variables (see Kandrup 1990; Perez & Aly 1996): the set of distribution functions is an infinite‐dimensional space. The total energy associated to a distribution function f can be written as …”

Section: Introductionmentioning

confidence: 99%

“…The distribution function depends both on the one‐particle energy E and on the squared one‐particle angular momentum . The most general result in this context was obtained by Perez & Aly (1996) and concerns linear stability for the restricted case of preserving perturbations, which includes radially symmetric ones. Non‐linear stability is assured for some classes of generalized polytropes for which f 0 ( E , L 2 ) = E k L 2 p with adapted values of k and p (see Rein 2002, and references within).…”

Section: Introductionmentioning

confidence: 99%

“…On the first hand, in , we present a general method for investigate instability of self‐gravitating systems. This approach couples a general mathematical result by Bloch et al (1994) which generalizes Lyapunov theory, and the symplectic approach of the stability problem of CBP system (see Bartolomew 1971; Kandrup 1990; Perez & Aly 1996) – it must be noted that Kandrup (1991) has already used this technique for non‐spherical systems without the complete mathematical background.…”

Section: Introductionmentioning

confidence: 99%

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Radial orbit instability as a dissipation-induced phenomenon

Maréchal¹,

Perez²

2010

Monthly Notices of the Royal Astronomical Society

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This paper is devoted to Radial Orbit Instability in the context of self-gravitating dynamical systems. We present this instability in the new frame of Dissipation-Induced Instability theory. This allows us to obtain a rather simple proof based on energetics arguments and to clarify the associated physical mechanism.Comment: 15 pages. Published in Monthly Notices of the RAS by the Royal Astronomical Society and Blackwell Publishing. Corrected for page style, typos, and added reference

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A Birman-Schwinger Principle in Galactic Dynamics

Kunze

2021

Progress in Mathematical Physics

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These are the (somewhat extended) lecture notes for four lectures delivered at the spring school during the thematic programme "Mathematical Perspectives of Gravitation beyond the Vacuum Regime" at ESI Vienna in February 2022. Contents 1 Introduction 2 The Birman-Schwinger principle in quantum mechanics 3 Galactic dynamics: The Vlasov-Poisson system 4 Spherically symmetric solutions 5 Steady state solutions 6 Action angle variables 7 Function spaces 8 Linearization 1 9 The Birman-Schwinger approach 21 10 An application 26 11 Open questions and further topics 30 Proof : See [28, Section 4.3.1]. If Hφ = (−∆+ V )φ = (−e)φ, then we define ψ = √ −V φ to obtainConversely, if B e ψ = ψ holds and if we put φ = (−∆ + e) −1 ( √ −V ψ), it follows thatand hence Hφ = (−∆ + V )φ = (−e)φ, which completes the argument. ✷Here are some facts:• The operators B e are non-negative Hilbert-Schmidt operators (if V decays sufficiently fast and n ≤ 3), and in particular they are compact.• Their eigenvalues can be ordered: λ 1 (e) ≥ λ 2 (e) ≥ . . . → 0 and the eigenvalue curves are decreasing in e, in that ẽ ≥ e implies that λ k (e) ≤ λ k (ẽ) for all k.

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Stability of spherical stellar systems -- I. Analytical results

Cited by 36 publications

References 1 publication

A new variational approach to the stability of gravitational systems

A new variational approach to the stability of gravitational systems

Radial orbit instability as a dissipation-induced phenomenon

A Birman-Schwinger Principle in Galactic Dynamics

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