Related papers: Reduction and Integrability
The integrability condition called shape invariance is shown to have an underlying algebraic structure and the associated Lie algebras are identified. These shape-invariance algebras transform the parameters of the potentials such as…
The lower invariance under a given arbitrary group of diffeomorphisms extends the notion of quasiconvexity. The non-commutativity of the group operation (the function composition) modifies the classical equivalence between lower…
In this paper, we seek to understand the behavior of dynamical systems that are perturbed by a parameter that changes discretely in time. If we impose certain conditions, we can study certain embedded systems within a hybrid system as…
Helicity plays a unique role as an integral invariant of a dynamical system. In this paper, the concept of helicity in the general setting of Hamiltonian dynamics is discussed. It is shown, through examples, how the conservation of overall…
The aim of this paper is to give a condition to topological conjugacy of invariant flows in an Lie group $G$ which its Lie algebra $\mathfrak{g}$ is associative algebra or semisimple. In fact, we show that if two dynamical system on $G$ are…
In this paper we explore the stabilization of closed invariant sets for passive systems, and present conditions under which a passivity-based feedback asymptotically stabilizes the goal set. Our results rely on novel reduction principles…
In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie…
Constrained Hamiltonian systems are investigated by using the Hamilton-Jacobi method. Integration of a set of equations of motion and the action function is discussed. It is shown that we have two types of integrable systems: a) ${\it…
We give a short and elementary introduction to Lie group methods. A selection of applications of Lie group integrators are discussed. Finally, a family of symplectic integrators on cotangent bundles of Lie groups is presented and the notion…
We consider a dynamical system consisting of subsystems indexed by a lattice. Each subsystem has one conserved degree of freedom ("energy") the rest being uniformly hyperbolic. The subsystems are weakly coupled together so that the sum of…
We develop a theory of gauge and dynamical equivalence for Lagrangian systems on Lie algebroids, also studying its relationship with Noether and non-Noether conserved quantities.
This paper is devoted to the study of mechanical systems subjected to external forces in the framework of symplectic geometry. We obtain a Noether's theorem for Lagrangian systems with external forces, among other results regarding…
In this paper we investigate a class of natural Hamiltonian systems with two degrees of freedom. The kinetic energy depends on coordinates but the system is homogeneous. Thanks to this property it admits, in a general case, a particular…
We extend some recent results about bounded invariant equivalence relations and invariant subgroups of definable groups: we show that type-definability and smoothness are equivalent conditions in a wider class of relations than heretofore…
One unusual property of dynamic systems, whose state is characterized by a set of scalar dynamic variables satisfying a system of differential equations of a general form, is considered. This property is related to the behavior of equations…
The integrability problem consists in finding the class of functions a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit an elementary or Liouvillian first…
Several problems in number theory when reformulated in terms of homogenous dynamics involve study of limiting distributions of translates of algebraically defined measures on orbits of reductive groups. The general non-divergence and…
Relations between parameter rigidity of locally free Lie group actions on closed manifolds and the 1st leafwise cohomology of the orbit foliations are discussed. Some computational results of the leafwise cohomology are included.
The concept of signature is a useful tool in the analysis of semicoherent systems with continuous and i.i.d. component lifetimes, especially for the comparison of different system designs and the computation of the system reliability. For…
This talk gives an introduction into the subject of Seiberg-Witten curves and their relation to integrable systems. We discuss some motivations and origins of this relation and consider explicit construction of various families of…