-body problem in : necessary conditions for a constant configuration measure.
Non-holonomic mechanical systems in jet bundles.
In this paper we present a geometrical formulation for Lagrangian systems subjected to non-holonomic constraints in terms of jet bundles. Cosymplectic geometry and almost product structures are used to obtained the constrained dynamics without using Lagrange multipliers method.
Non-integrability of certain Hamiltonian systems. Applications of the Morales-Ramis differential Galois extension of Ziglin theory
The aim of this paper is to present two examples of non academic Hamiltonian systems for which the Morales-Ramis theory can be applied effectively. First, we investigate the Gross-Neveu system with n degrees of freedom. Till now it has been proved that this system is not integrable for n = 3. We give a simple proof that it is not completely integrable for an arbitrary n ≥ 3. Our second example is a natural generalisation of the Jacobi problem of a material point moving on an ellipsoid. We formulate...
Nonsphericity of the Moon and near Sun-synchronous polar lunar orbits.
Note sur le problème des trois corps.
Oeuvres de Lagrange Tome 4 [Book]
Oeuvres de Lagrange Tome 6 [Book]
On Carnot's theorem in time dependent impulsive mechanics.
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 central configurations.
On control problems of minimum time for Lagrangian systems similar to a swing. I. Convexity criteria for sets
One establishes some convexity criteria for sets in . They will be applied in a further Note to treat the existence of solutions to minimum time problems for certain Lagrangian systems referred to two coordinates, one of which is used as a control. These problems regard the swing or the ski.
On control problems of minimum time for Lagrangian systems similar to a swing. II Application of convexity criteria to certain minimum time problems
This Note is the Part II of a previous Note with the same title. One refers to holonomic systems with two degrees of freedom, where the part can schemetize a swing or a pair of skis and schemetizes whom uses . The behaviour of is characterized by a coordinate used as a control. Frictions and air resistance are neglected. One considers on minimum time problems and one is interested in the existence of solutions. To this aim one determines a certain structural condition which implies...
On D’Alembert’s Principle
A formulation of the D’Alembert principle as the orthogonal projection of the acceleration onto an affine plane determined by nonlinear nonholonomic constraints is given. Consequences of this formulation for the equations of motion are discussed in the context of several examples, together with the attendant singular reduction theory.
On existence and uniqueness of the solution of the equation of motion for constrained mechanical systems.
On integration factor of a dynamical system.
On integration of the differential equation of central motion, I
On measure solutions to the zero-pressure gas model and their uniqueness
On measure solutions to the Zero-pressure gas model and their uniqueness
The system of zero-pressure gas dynamics conservation laws describes the dynamics of free particles sticking under collision while mass and momentum are conserved. The existence of such solutions was established some time ago. Here we report a uniqueness result that uses the Oleinik entropy condition and a cohesion condition. Both of these conditions are automatically satisfied by solutions obtained in previous existence results. Important tools in the proof of uniqueness are regularizations, generalized...
On non-existence of periodic solutions of an important differential equation
On submanifolds and quotients of Poisson and Jacobi manifolds
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 symmetries and constants of motion in Hamiltonian systems with nonholonomic constraints