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On motions with bursting characters for Lagrangian mechanical systems with a scalar control. I. Existence of a wide class of Lagrangian systems capable of motions with bursting characters

Aldo Bressan, Marco Favretti (1991)

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

In this Note (which will be followed by a second) we consider a Lagrangian system Σ (possibly without any Lagrangian function) referred to N + 1 coordinates q 1 ⁢ ⋯ , q N , u , with u to be used as a control, and precisely to add to Σ a frictionless constraint of the type u = u ⁢ t . Let Σ 's (frictionless) constraints be represented by the manifold V t generally moving in Hertz's space. We also consider an instant d (to be used for certain limit discontinuity-properties), a point q ¯ , u ¯ of V d , a value p ¯ for Σ 's momentum conjugate...

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 some completions of the space of hamiltonian maps

Vincent Humilière (2008)

Bulletin de la Société Mathématique de France

In one of his papers, C. Viterbo defined a distance on the set of Hamiltonian diffeomorphisms of ℝ 2 n endowed with the standard symplectic form ω 0 = d p ∧ d q . We study the completions of this space for the topology induced by Viterbo’s distance and some others derived from it, we study their different inclusions and give some of their properties. In particular, we give a convergence criterion for these distances that allows us to prove that the completions contain non-ordinary elements, as for example, discontinuous...

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 analytic non-integrability of the Rattleback problem

H. R. Dullin, A.V. Tsygvintsev (2008)

Annales de la faculté des sciences de Toulouse Mathématiques

We establish the analytic non-integrability of the nonholonomic ellipsoidal rattleback model for a large class of parameter values. Our approach is based on the study of the monodromy group of the normal variational equations around a particular orbit. The imbedding of the equations of the heavy rigid body into the rattleback model is discussed.

On the application of control theory to certain problems for Lagrangian systems, and hyper-impulsive motion for these. I. Some general mathematical considerations on controllizable parameters

Aldo Bressan (1988)

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

In applying control (or feedback) theory to (mechanic) Lagrangian systems, so far forces have been generally used as values of the control u ⁢ ( ⋅ ) . However these values are those of a Lagrangian co-ordinate in various interesting problems with a scalar control u = u ⁢ ( ⋅ ) , where this control is carried out physically by adding some frictionless constraints. This pushed the author to consider a typical Lagrangian system Σ , referred to a system χ of Lagrangian co-ordinates, and to try and write some handy conditions,...

On the classical non-integrability of the Hamiltonian system for hydrogen atoms in crossed electric and magnetic fields

Robert Gębarowski (2011)

Banach Center Publications

Hydrogen atoms placed in external fields serve as a paradigm of a strongly coupled multidimensional Hamiltonian system. This system has been already very extensively studied, using experimental measurements and a wealth of theoretical methods. In this work, we apply the Morales-Ramis theory of non-integrability of Hamiltonian systems to the case of the hydrogen atom in perpendicular (crossed) static electric and magnetic uniform fields.

On the determination of the potential function from given orbits

L. Alboul, J. Mencía, R. Ramírez, N. Sadovskaia (2008)

Czechoslovak Mathematical Journal

The paper deals with the problem of finding the field of force that generates a given ( N - 1 )-parametric family of orbits for a mechanical system with N degrees of freedom. This problem is usually referred to as the inverse problem of dynamics. We study this problem in relation to the problems of celestial mechanics. We state and solve a generalization of the Dainelli and Joukovski problem and propose a new approach to solve the inverse Suslov’s problem. We apply the obtained results to generalize the...

On the differential equations of the classical and relativistic dynamics for certain generalised Lagrangian functions

Antonio Pignedoli (1987)

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

One studies the differential equations of the movement of certain classical and relativistic systems for some special Lagrangian functions. One considers particularly the case in which the problem presents cyclic coordinates. Some electrodynamical applications are studied.

On the inverse problem of the calculus of variations for ordinary differential equations

Olga Krupková (1993)

Mathematica Bohemica

Lepagean 2-form as a globally defined, closed counterpart of higher-order variational equations on fibered manifolds over one-dimensional bases is introduced, and elementary proofs of the basic theorems concerning the inverse problem of the calculus of variations, based on the notion of Lepagean 2-form and its properties, are given.

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