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On a deformed version of the two-disk dynamo system

Cristian Lăzureanu, Camelia Petrişor, Ciprian Hedrea (2021)

Applications of Mathematics

We give some deformations of the Rikitake two-disk dynamo system. Particularly, we consider an integrable deformation of an integrable version of the Rikitake system. The deformed system is a three-dimensional Hamilton-Poisson system. We present two Lie-Poisson structures and also symplectic realizations. Furthermore, we give a prequantization result of one of the Poisson manifold. We study the stability of the equilibrium states and we prove the existence of periodic orbits. We analyze some properties...

On Carnot's theorem in time dependent impulsive mechanics.

Stefano Pasquero (2005)

Extracta Mathematicae

We show that the validity of the Carnot's theorem about the kinetic energy balance for a mechanical system subject to an inert impulsive kinetic constraint, once correctly framed in the time dependent geometric environment for Impulsive Mechanics given by the left and right jet bundles of the space-time bundle N, is strictly related to the frame of reference used to describe the system and then it is not an intrinsic property of the mechanical system itself. We analyze in details the class of frames...

On Liouville forms

Paulette Libermann (2000)

Banach Center Publications

We give different notions of Liouville forms, generalized Liouville forms and vertical Liouville forms with respect to a locally trivial fibration π:E → M. These notions are linked with those of semi-basic forms and vertical forms. We study the infinitesimal automorphisms of these forms; we also investigate the relations with momentum maps.

On singularities of Hamiltonian mappings

Takuo Fukuda, Stanisław Janeczko (2008)

Banach Center Publications

The notion of an implicit Hamiltonian system-an isotropic mapping H: M → (TM,ω̇) into the tangent bundle endowed with the symplectic structure defined by canonical morphism between tangent and cotangent bundles of M-is studied. The corank one singularities of such systems are classified. Their transversality conditions in the 1-jet space of isotropic mappings are described and the corresponding symplectically invariant algebras of Hamiltonian generating functions are calculated.

On submanifolds and quotients of Poisson and Jacobi manifolds

Charles-Michel Marle (2000)

Banach Center Publications

We obtain conditions under which a submanifold of a Poisson manifold has an induced Poisson structure, which encompass both the Poisson submanifolds of A. Weinstein [21] and the Poisson structures on the phase space of a mechanical system with kinematic constraints of Van der Schaft and Maschke [20]. Generalizations of these results for submanifolds of a Jacobi manifold are briefly sketched.

On the Isoenergetical Non-Degeneracy of the Problem of two Centers of Gravitation

Dragnev, Dragomir (1997)

Serdica Mathematical Journal

* Partialy supported by contract MM 523/95 with Ministry of Science and Technologies of Republic of Bulgaria.For the system describing the motion of a moss point under the action of two static gravity centers (with equal masses), we find a subset of the set of the regular values of the energy and momentum, where the condition of isoenergetical non-degeneracy is fulfilled.

On the multiplicity of brake orbits and homoclinics in Riemannian manifolds

Roberto Giambò, Fabio Giannoni, Paolo Piccione (2005)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let M , g be a complete Riemannian manifold, Ω ⊂ M an open subset whose closure is diffeomorphic to an annulus. If ∂ ⁡ Ω is smooth and it satisfies a strong concavity assumption, then it is possible to prove that there are at least two geometrically distinct geodesics in Ω ¯ = Ω ⁢ ⋃ ∂ ⁡ Ω starting orthogonally to one connected component of ∂ ⁡ Ω and arriving orthogonally onto the other one. The results given in [5] allow to obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating...

Overview of the differential Galois integrability conditions for non-homogeneous potentials

Andrzej J. Maciejewski, Maria Przybylska (2011)

Banach Center Publications

We report our recent results concerning integrability of Hamiltonian systems governed by Hamilton’s function of the form H = 1 / 2 ∑ i = 1 n p ² i + V ( q ) , where the potential V is a finite sum of homogeneous components. In this paper we show how to find, in the differential Galois framework, computable necessary conditions for the integrability of such systems. Our main result concerns potentials of the form V = V k + V K , where V k and V K are homogeneous functions of integer degrees k and K > k, respectively. We present examples of integrable...

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