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On the alpine ski with dry friction and air resistance. Some optimization problems for it

Aldo Bressan (1999)

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

In the present work, divided in three parts, one considers a real skis-skier system, Σ R , descending along a straight-line l with constant dry friction; and one schematizes it by a holonomic system Σ = A ∪ U , having any number n ≥ 4 of degrees of freedom and subjected to (non-ideal) constraints, partly one-sided. Thus, e.g., jumps and also «steps made with sliding skis» can be schematized by Σ . Among the n Lagrangian coordinates for Σ two are the Cartesian coordinates ξ and η of its center of mass, C , relative...

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 derivation of a quantum Boltzmann equation from the periodic Von-Neumann equation

François Castella (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We present the semi-conductor Boltzmann equation, which is time-reversible, and indicate that it can be formally derived by considering the large time and small perturbing potential limit in the Von-Neumann equation (time-reversible). We then rigorously compute the corresponding asymptotics in the case of the Von-Neumann equation on the Torus. We show that the limiting equation we obtain does not coincide with the physically realistic model. The former is indeed an equation of Boltzmann type, yet...

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 inverse variational problem in nonholonomic mechanics

Olga Rossi, Jana Musilová (2012)

Communications in Mathematics

The inverse problem of the calculus of variations in a nonholonomic setting is studied. The concept of constraint variationality is introduced on the basis of a recently discovered nonholonomic variational principle. Variational properties of first order mechanical systems with general nonholonomic constraints are studied. It is shown that constraint variationality is equivalent with the existence of a closed representative in the class of 2-forms determining the nonholonomic system. Together with...

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 mobility and efficiency of mechanical systems

Gershon Wolansky (2007)

ESAIM: Control, Optimisation and Calculus of Variations

It is shown that self-locomotion is possible for a body in Euclidian space, provided its dynamics corresponds to a non-quadratic Hamiltonian, and that the body contains at least 3 particles. The efficiency of the driver of such a system is defined. The existence of an optimal (most efficient) driver is proved.


On the Singularities of the Newtonian two dimensional N-body Problem

Carlo Marchioro, Mario Pulvirenti (1983)

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

Si considera un sistema bidimensionale di N particelle interagenti tramite un potenziale di Newton o di Coulomb e si mostra che l’insieme delle condizioni iniziali che in un tempo finito possono condurre a delle singolarità possiede misura di Lebesgue nulla.

On weak solutions to the Lagrange-d'Alembert equation

Dmitry Treschev, Oleg Zubelevich (2013)

Applicationes Mathematicae

We consider nonholonomic systems with collisions and propose a concept of weak solutions to Lagrange-d'Alembert equations. Using this concept we describe the dynamics of collisions. Collisions of a rotating ball and a rough floor are considered.

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