Hamiltonization of the generalized Veselova LR system

Fedorov, Yu. N.; Jovanović, Božidar

doi:10.1134/s1560354709040066

Cited by 22 publications

(46 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In Section 4.2 we prove that the problem considered by Fedorov and Jovanović [19,20] is φsimple (Theorem 4.3) so the Chaplygin Hamiltonisation of the problem may be understood within our geometric framework.…”

Section: Chaplygin Hamiltonisation Of the Multi-dimensional Veselova mentioning

confidence: 94%

“…Moreover, we express sufficient conditions for measure preservation and Hamiltonisation via Chaplygin's reducing multiplier method in terms of the properties of this tensor. The theory is used to give a new proof of the remarkable Hamiltonisation of the multi-dimensional Veselova system obtained by Fedorov and Jovanović in [19,20]. where g denotes the Lie algebra of G and g ⋅ q the tangent space to the G-orbit through q.These systems were first considered by Chaplygin and Hamel around the year 1900, and their geometric features have since been investigated by a number of authors e.g. [1,30,47,36,6,11,19,13,10,27,3,21] and references therein.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

The geometry of nonholonomic Chaplygin systems revisited

García-Naranjo

Marrero

2020

Nonlinearity

View full text Add to dashboard Cite

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 kinetic energy metric. We show how this tensor may be naturally used to derive an almost symplectic description of the reduced dynamics. Moreover, we express sufficient conditions for measure preservation and Hamiltonisation via Chaplygin's reducing multiplier method in terms of the properties of this tensor. The theory is used to give a new proof of the remarkable Hamiltonisation of the multi-dimensional Veselova system obtained by Fedorov and Jovanović in [19,20]. where g denotes the Lie algebra of G and g ⋅ q the tangent space to the G-orbit through q.These systems were first considered by Chaplygin and Hamel around the year 1900, and their geometric features have since been investigated by a number of authors e.g. [1,30,47,36,6,11,19,13,10,27,3,21] and references therein. Their key feature is that the reduced equations of motion may be formulated as an unconstrained, forced mechanical system on the shape space S = Q G. The dimension r of S is termed the number of degrees of freedom and, because of (1.1), it coincides with the rank of the constraint distribution D (in particular, the non-integrability of D implies that r ≥ 2).Our contribution to the subject is to highlight the relevance of a tensor field T defined on S, that we term the gyroscopic tensor, in the structure of the reduced equations of motion and their properties. Moreover, using this tensor, we are able to single out a very special class of systems, that we term φ -simple, which possess an invariant measure and allow a Hamiltonisation, and for which there exist non-trivial examples. The gyroscopic tensorThe gyroscopic tensor T is introduced in Definition 3.3. It is a (1,2) skew-symmetric tensor field on the shape space S, that to a pair of vector fields Y,Z on S, assigns a third vector field T (Y,Z) on S. The assignment is done in a manner that measures the interplay between the nonintegrability of the noholonomic constraint distribution and the kinetic energy of the system. In particular, in the case of holonomic constraints, where the constraint distribution is integrable, we have T = 0. We mention that there is a close relation between T and the geometric formulation of nonholonomic systems in terms of linear almost Poisson brackets on vector bundles [26,40] (see also [21]).Although the tensor T appears in the previous works of Koiller [36] and Cantrijn et al. [11] (with an alternative definition than the one that we present here), its dynamical relevance had not been fully appreciated until the recent work García-Naranjo [24] where sufficient conditions for Hamiltonisation were given in terms of the coordinate representation of T . This work continues the research started in [24] by providing a coordinate-free definition of the gyroscopic tensor (Definition 3.3), and studying in depth its role in the al...

show abstract

Section: Chaplygin Hamiltonisation Of the Multi-dimensional Veselova mentioning

confidence: 94%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

The geometry of nonholonomic Chaplygin systems revisited

García-Naranjo

Marrero

2020

Nonlinearity

View full text Add to dashboard Cite

show abstract

“…The study of multi-dimensional systems in nonholonomic mechanics goes back to Fedorov and Kozlov [15], and has received wide attention as a source of interesting examples for integrability, Hamiltonisation and other types of dynamical features [37,26,16,17,28,29,13,14,19]. Our analysis of the multi-dimensional rubber Routh sphere contributes to enlarge this family of examples.…”

Section: The Multi-dimensional Rubber Routh Spherementioning

confidence: 96%

Hamiltonisation, measure preservation and first integrals of the multi-dimensional rubber Routh sphere

García-Naranjo

2019

Theor appl mech (Belgr)

View full text Add to dashboard Cite

We consider the multi-dimensional generalisation of the problem of a sphere, with axi-symmetric mass distribution, that rolls without slipping or spinning over a plane. Using recent results from García-Naranjo [21] and García-Naranjo and Marrero [22], we show that the reduced equations of motion possess an invariant measure and may be represented in Hamiltonian form by Chaplygin's reducing multiplier method. We also prove a general result on the existence of first integrals for certain Hamiltonisable Chaplygin systems with internal symmetries that is used to determine conserved quantities of the problem. * This research was made possible by a Georg Forster Experienced Researcher Fellowship from the Alexander von Humboldt Foundation that funded a research stay of the author at TU Berlin. 1 the shape space is the quotient manifold S = Q G where Q is the configuration manifold of the system and G is the underlying symmetry group of the Chaplygin system. 2 throughout the paper we denote by Y [ f ] the action of the vector field Y on the scalar function f .

show abstract

“…A large variety of examples can be found in Neimark and Fuffaev book [112]. It should be noticed that, related with these problems, there is a relevant research connected with differential and algebraic geometry: Agrachev [2], Fatima Leites and co-workers [63], Jurdjevic [71,72],… -Another important source of examples, coming from the study of rigid bodies, is the study of nonholonomic systems on Lie groups: Veselov and Veselova [143,144], Fedorov and co-workers [48,49], … -In recent times, the so-called nonholonomic toys have deserved a lot of attention from the scholars: the wowblestone, ratleback or celtic stone (see the papers by Bondi [20] and Tokieda and Moffatt [109]; the skateboard (see Kuleshov [82], Kremnev and Kuleshov [81]); the snakeboard (see Ferraro, Kobilarov, de León, Martín de Diego and Marrero [86,74]). …”

Section: Nonholonomic Examples and Devicesmentioning

confidence: 99%

A historical review on nonholomic mechanics

León

2011

RACSAM

View full text Add to dashboard Cite

The aim of this paper is to present a short historical/scientific review on nonholonomic mechanics, with special emphasis on the latest developments. Indeed, the use of differential geometric tools has permitted in the last 25 years a fast and unsuspected advance in the theory, particularly in a better understanding of symmetries and reduction, Hamilton-Jacobi theory and integrability characterizations, and the construction of suitable geometric integrators. The last part of the paper is devoted to discuss the latest results in Hamilton-Jacobi theory for nonholonomic dynamics using our own approach.

show abstract

Hamiltonization of the generalized Veselova LR system

Cited by 22 publications

References 20 publications

The geometry of nonholonomic Chaplygin systems revisited

The geometry of nonholonomic Chaplygin systems revisited

Hamiltonisation, measure preservation and first integrals of the multi-dimensional rubber Routh sphere

A historical review on nonholomic mechanics

Contact Info

Product

Resources

About