The global control over asymptotical invariants of linear systems in small dimensions

Козлов, А. А.

doi:10.20537/vm080219

Cited by 2 publications

(7 citation statements)

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“…Equation (6) for Hamiltonian systems with n = 2m was derived by Kozlov [4][5][6], who called it the Lamb equation. It is in this case (n = 2m) that Eq.…”

Section: The Hydrodynamic Substitution In the Hamiltonian And Non Hammentioning

confidence: 99%

“…The hydrodynamic substitution, which is well known in the theory of the Vlasov equation [1][2][3], has recently been applied to the Liouville equation and Hamiltonian mechanics [4][5][6][7][8]. In [4][5][6], Kozlov out lined the simplest derivation of the Hamilton-Jacobi (HJ) equation, and the hydrodynamic substitution simply related this derivation to the Liouville equation [7,8].…”

Section: Introductionmentioning

confidence: 99%

“…In [4][5][6], Kozlov out lined the simplest derivation of the Hamilton-Jacobi (HJ) equation, and the hydrodynamic substitution simply related this derivation to the Liouville equation [7,8]. The hydrodynamic substitution also solves the interesting geometric problem of how a surface of any dimension subject to an arbitrary system of nonlinear ordinary differential equations moves in Euler coordi nates (in Lagrangian coordinates, the answer is obvi ous).…”

Section: Introductionmentioning

confidence: 99%

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The Hamilton-Jacobi method in the non-Hamiltonian situation and the hydrodynamic substitution

Vedenyapin

Фимин

2015

Dokl. Math.

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“…Equation (6) for Hamiltonian systems with n = 2m was derived by Kozlov [4][5][6], who called it the Lamb equation. It is in this case (n = 2m) that Eq.…”

Section: The Hydrodynamic Substitution In the Hamiltonian And Non Hammentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Hamilton-Jacobi method in the non-Hamiltonian situation and the hydrodynamic substitution

Vedenyapin

Фимин

2015

Dokl. Math.

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“…In 1976 V. V. Kozlov in his paper [1] (see also [2,3]) proved a theorem which gives sufficient conditions of the nonexistence, for the Hamiltonian system, of a first integral analytic in canonical variables and independent of the Hamiltonian function H. Below we give a statement of the problem using the notations from [1] and a formulation of the corresponding theorem.…”

Section: Introduction V V Kozlov's Theorem On the Nonexistence Of An ...mentioning

confidence: 98%

“…V. V. Kozlov's Theorem 1 was successfully applied to prove the nonexistence of an additional first integral in the plane circular restricted three-body problem [1][2][3], to study the integrability of the problem of motion about a fixed point of a dynamically symmetric rigid body with the center of mass lying in the equatorial plane of the ellipsoid of inertia [1,3,6], to prove the nonexistence of an additional integral in the problem of the motion of a heavy double plane pendulum [6][7][8], to obtain necessary conditions for the existence of an additional first integral in the problem of the motion of a dynamically symmetric ellipsoid on a smooth horizontal plane [9], and to study nonintegrability of the Kirchhoff equations of motion of a rigid body in a fluid [10,11].…”

Section: Introduction V V Kozlov's Theorem On the Nonexistence Of An ...mentioning

confidence: 99%

Nonintegrability of the Problem of the Motion of an Ellipsoidal Body with a Fixed Point in a Flow of Particles

Gadzhiev¹,

Kuleshov²

2022

Nelin. Dinam.

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The problem of the motion, in the free molecular flow of particles, of a rigid body with a fixed point bounded by the surface of an ellipsoid of revolution is considered. This problem is similar in many aspects to the classical problem of the motion of a heavy rigid body about a fixed point. In particular, this problem possesses the integrable cases corresponding to the classical Euler – Poinsot, Lagrange and Hess cases of integrability of the equations of motion of a heavy rigid body with a fixed point. A natural question arises about the existence of analogues of other integrable cases that exist in the problem of motion of a heavy rigid body with a fixed point (Kovalevskaya case, Goryachev –Chaplygin case, etc) for the system considered. Using the standard Euler angles as generalized coordinates, the Hamiltonian function of the system is derived. Equations of motion of the body in the flow of particles are presented in Hamiltonian form. Using the theorem on the Liouville-type nonintegrability of Hamiltonian systems near elliptic equilibrium positions, which has been proved by V. V.Kozlov, necessary conditions for the existence in the problem under consideration of an additional analytic first integral independent of the energy integral are presented. We have proved that the necessary conditions obtained are not fulfilled for a rigid body with a mass distribution corresponding to the classical Kovalevskaya integrable case in the problem of the motion of a heavy rigid body with a fixed point. Thus, we can conclude that this system does not possess an integrable case similar to the Kovalevskaya integrable case in the problem of the motion of a heavy rigid body with a fixed point.

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The global control over asymptotical invariants of linear systems in small dimensions

Cited by 2 publications

References 0 publications

The Hamilton-Jacobi method in the non-Hamiltonian situation and the hydrodynamic substitution

The Hamilton-Jacobi method in the non-Hamiltonian situation and the hydrodynamic substitution

Nonintegrability of the Problem of the Motion of an Ellipsoidal Body with a Fixed Point in a Flow of Particles

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