相关论文: Geometrisation of Chaplygin's reducing multiplier …
In this work, we conduct a systematic study of Hamiltonian and quasi-Hamiltonian systems within the framework of nondecomposable generalized Poisson geometry. Our focus lies on the interplay between the algebraic structure of…
In purely non-dissipative systems, Lagrangian and Hamiltonian reduction have proven to be powerful tools for deriving physical models with exact conservation laws. We have discovered a hint that an analogous reduction method exists also for…
We discuss the Poisson structures, Lax matrices, $r$-matrices, bi-hamiltonian structures, the variables of separation and other attributes of the modern theory of dynamical systems in application to the integrable Euler top and to the…
We show that some modern geometric methods of Hamiltonian dynamics can be directly applied to the nonholonomic Heisenberg type systems. As an example we present characteristic Killing tensors, compatible Poisson brackets, Lax matrices and…
We introduce and study the dynamics of Chebyshev polynomials on $d>2$ real intervals. We define isoharmonic deformations as a natural generalization of the Chebyshev dynamics. This dynamics is associated with a novel class of constrained…
We show that, under suitable conditions, finite-dimensional systems describing invariant solutions of partial differential equations (PDEs) inherit local Hamiltonian operators through the mechanism of invariant reduction, which applies…
In this paper we study the relation between conserved quantities of nonholonomic systems and the hamiltonization problem employing the geometric methods of [1,3]. We illustrate the theory with classical examples describing the dynamics of…
We discuss linear in momenta Poisson structure for the generalized nonholonomic Chaplygin sphere problem and prove that it is non-trivial deformation of the canonical Poisson structure on e^*(3).
A consistent framework has been put forward to quantize the isentropic, compressible and inviscid fluid model in the Hamiltonian framework, using the Clebsch parameterization. The naive quantization is hampered by the non-canonical (in…
In this paper the well-known Dubrovin-Novikov problem posed as long ago as 1984 in connection with the Hamiltonian theory of systems of hydrodynamic type, namely, the classification problem for multidimensional Poisson brackets of…
The rolling of a dynamically balanced ball on a horizontal rough table without slipping was described by Chaplygin using Abel quadratures. We discuss integrable discretizations and deformations of this nonholonomic system using the same…
Via compression ([11, 7]) we write the $n$-dimensional Chaplygin sphere system as an almost Hamiltonian system on $T^*SO(n)$ with internal symmetry group $SO(n-1)$. We show how this symmetry group can be factored out, and pass to the fully…
We present a geometric construction of irreversible dynamics on Poisson manifolds that satisfies the axioms of metriplectic mechanics and the GENERIC framework. Our approach relies solely on the underlying Poisson structure and its…
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 paper we explore the nonholonomic Lagrangian setting of mechanical systems in local coordinates on finite-dimensional configuration manifolds. We prove existence and uniqueness of solutions by reducing the basic equations of motion…
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,…
This paper studies hamiltonization of nonholonomic systems using geometric tools. By making use of symmetries and suitable first integrals of the system, we explicitly define a global 2-form for which the gauge transformed nonholonomic…
We introduce and study the Chaplygin systems with gyroscopic forces. This natural class of nonholonomic systems has not been treated before. We put a special emphasis on the important subclass of such systems with magnetic forces. The…
Borisov, Mamaev and Kilin have recently found certain Poisson structures with respect to which the reduced and rescaled systems of certain non-holonomic problems, involving rolling bodies without slipping, become Hamiltonian, the…
We discuss a non-Hamiltonian vector field appearing in consideration of a partial motion of the Chaplygin ball rolling on a horizontal plane which rotates with constant angular velocity. In two partial cases this vector field is expressed…