Related papers: On the geometry of generalized Chaplygin systems
We apply methods of dynamical systems to study the behaviour of universe dominated by the generalized Chaplygin gas. We reduce the dynamics to a 2-dimensional Hamiltonian system and study its behaviour for various ranges of parameters. The…
A generalisation of Chaplygin's Reducing Multiplier Theorem is given by providing sufficient conditions for the Hamiltonisation of Chaplygin nonholonomic systems with an arbitrary number $r$ of degrees of freedom via Chaplygin's multiplier…
In this paper, we prove that the trajectories of unreduced $\phi$-simple Chaplygin kinetic systems are reparametrizations of horizontal geodesics with respect to a modified Riemannian metric. Furthermore, our proof is constructive and these…
We study relations between vakonomically and nonholonomically constrained Lagrangian dynamics for the same set of linear constraints. The basic idea is to compare both situations at the level of variational principles, not equations of…
A gauge independent method of obtaining the reduced space of constrained dynamical systems is discussed in a purely lagrangian formalism. Implications of gauge fixing are also considered.
We consider nonholonomic Chaplygin systems and associate to them a $(1,2)$ tensor field on the shape space, that we term the gyroscopic tensor, and that measures the interplay between the non-integrability of the constraint distribution and…
The reduction of dynamical systems has a rich history, with many important applications related to stability, control and verification. Reduction of nonlinear systems is typically performed in an exact manner - as is the case with…
For a Lagrangian system with nonholonomic constraints, we construct extensions of the equations of motion to sets of second-order ordinary differential equations. In the case of a purely kinetic Lagrangian, we investigate the conditions…
We consider the stability of synchronized states (including equilibrium point, periodic orbit or chaotic attractor) in arbitrarily coupled dynamical systems (maps or ordinary differential equations). We develop a general approach, based on…
We develop the reducing multiplier theory for a special class of nonholonomic dynamical systems and show that the non-linear Poisson brackets naturally obtained in the framework of this approach are all isomorphic to the Lie-Poisson…
This paper presents a reduction procedure for nonholonomic systems admitting suitable types of symmetries and conserved quantities. The full procedure contains two steps. The first (simple) step results in a Chaplygin system, described by…
In this article, we generalize the theory of discrete Lagrangian mechanics and variational integrators in two principal directions. First, we show that Lagrangian submanifolds of symplectic groupoids give rise to discrete dynamical systems,…
Actions of Lie groups on presymplectic manifolds are analyzed, introducing the suitable comomentum and momentum maps. The subsequent theory of reduction of presymplectic dynamical systems with symmetry is studied. In this way, we give a…
We study the relationship between the partially synchronous state and the coupling structure in general dynamical systems. Our results show that, on the contrary to the widely accepted concept, topological symmetry in a coupling structure…
We consider a class of dynamical systems on a Lie group $G$ with a left-invariant metric and right-invariant nonholonomic constraints (so called LR systems) and show that, under a generic condition on the constraints, such systems can be…
We show that the Suslov nonholonomic rigid body problem can be regarded almost everywhere as a generalized Chaplygin system. Furthermore, this provides a new example of a multidimensional nonholonomic system which can be reduced to a…
The aim of this work is to study the geometry underlying mechanics and its application to describe autonomous and nonautonomous conservative dynamical systems of different types; as well as dissipative dynamical systems. We use different…
To establish a relation between two approaches to the construction of Riemann k-wave solutions of hydrodynamic-type systems, namely the symmetry reduction method and the generalized method of characteristics.
This paper develops a geometric approach to the theory of integrability by hydrodynamic reductions to establish an equivalence, for a large class of quasilinear systems, between hydrodynamic integrability and the existence of nets…
Projective geodesic extensions are reparametrizations of the trajectories of a nonholonomic mechanical system (with only a kinetic energy Lagrangian), in such a way that they can be interpreted as part of the geodesics of a Riemannian…