Conservation of energy and momenta in nonholonomic systems with affine constraints

Fassò, Francesco; Sansonetto, Nicola

doi:10.1134/s1560354715040048

Cited by 26 publications

(36 citation statements)

References 29 publications

(76 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The following consequence of Proposition 2 extends a result in [16] and is a possible statement of a 'Nonholonomic Noether theorem' for nonholonomic systems with affine constraints:…”

Section: Conservation Of Momenta Of Vector Fieldssupporting

confidence: 64%

See 1 more Smart Citation

Moving energies as first integrals of nonholonomic systems with affine constraints

2018

Self Cite

View full text Add to dashboard Cite

In nonholonomic mechanical systems with constraints that are affine (linear nonhomogeneous) functions of the velocities, the energy is typically not a first integral. It was shown in [17] that, nevertheless, there exist modifications of the energy, called there moving energies, which under suitable conditions are first integrals. The first goal of this paper is to study the properties of these functions and the conditions that lead to their conservation. In particular, we enlarge the class of moving energies considered in [17]. The second goal of the paper is to demonstrate the relevance of moving energies in nonholonomic mechanics. We show that certain first integrals of some well known systems (the affine Veselova and LR systems), which had been detected on a case-by-case way, are instances of moving energies. Moreover, we determine conserved moving energies for a class of affine systems on Lie groups that include the LR systems, for a heavy convex rigid body that rolls without slipping on a uniformly rotating plane, and for an n-dimensional generalization of the Chaplygin sphere problem to a uniformly rotating hyperplane.1 Reference [6], that refers to the moving energy as Jacobi integral (see also Section 2.3), claims that this mechanism can be extended to less symmetrical situations. It is unclear to us to which extent this goal can be achieved: without symmetry, the moving energy exists but is usually time-dependent. For more precise comments see footnote nr. 2 of [16]. We note that while in [17] and in the present article the moving energy is regarded as a first integral of the original system, and is therefore written in the original coordinates, in [6] it is written in the new, time-dependent coordinates.2 While the writing of this article was almost completed, we were informed of the existence of the very recent article [27]. Following the approach in [26], this article considers moving energies from the point of view of Noether symmetries for time-dependent systems-calling them Noether integrals-and generalizes to this context some of the results of [17]; in particular, it proves a statement analogous to Corollary 5.

show abstract

“…The following consequence of Proposition 2 extends a result in [16] and is a possible statement of a 'Nonholonomic Noether theorem' for nonholonomic systems with affine constraints:…”

Section: Conservation Of Momenta Of Vector Fieldssupporting

confidence: 64%

“…, k). For details see [16]; in the case of linear constraints, these or analogue expressions are given in [1,2,14]. We note that the restriction of R to M is independent of the arbitrariness that affects the choices of the vector field Z, of the matrix S and of the vector s, see [16].…”

Section: The Settingmentioning

confidence: 99%

Moving energies as first integrals of nonholonomic systems with affine constraints

2018

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the papers [26,27], which have appeared recently, Fassó explores conditions under which a system with inhomogeneous constraints (1.4) admits the energy integral (1.2).…”

Section: Remarkmentioning

confidence: 99%

“…We also obtain general conditions for existence of the Jacobi integral for inhomogeneous constraints, and in the (inhomogeneous) Suslov problem we find conditions for system parameters under which this integral exists (but is hidden now). In conclusion, we mention the work [27], which was published after we had sent our paper for printing and in which the conditions for existence of an energy integral are also discussed.…”

Section: Introductionmentioning

confidence: 95%

The jacobi integral in nonholonomic mechanics

2015

View full text Add to dashboard Cite

In this paper we discuss conditions for the existence of the Jacobi integral (that generalizes energy) in systems with inhomogeneous and nonholonomic constraints. As an example, we consider in detail the problem of motion of the Chaplygin sleigh on a rotating plane and the motion of a dynamically symmetric ball on a uniformly rotating surface. In addition, we discuss illustrative mechanical models based on the motion of a homogeneous ball on a rotating table and on the Beltrami surface.MSC2010 numbers: 70F25, 37J60, 70E18

show abstract

“…In Section 3 we apply Theorem 2.2 and obtain the main results: a general momentum equation and Noether theorem for time-dependent nonholonomic systems subjected to nonconservative forces (Theorems 3.1, 3.2). In particular, when we deal with symmetries that are prolongation of time-dependent vector fields on the configuration space, we get a time-dependent variant of the so called gauge symmetries studied in [2,3,14,16] (see Corollary 4.1) and the moving energy integral given in [16,17] (see Corollary 4.2).…”

Section: Introductionmentioning

confidence: 99%

Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems

Jovanović¹

2018

Journal of Geometric Mechanics

View full text Add to dashboard Cite

We consider Noether symmetries of the equations defined by the sections of characteristic line bundles of nondegenerate 1-forms and of the associated perturbed systems. It appears that this framework can be used for time-dependent systems with constraints and nonconservative forces, allowing a quite simple and transparent formulation of the momentum equation and the Noether theorem in their general forms.

show abstract

Conservation of energy and momenta in nonholonomic systems with affine constraints

Cited by 26 publications

References 29 publications

Moving energies as first integrals of nonholonomic systems with affine constraints

Moving energies as first integrals of nonholonomic systems with affine constraints

The jacobi integral in nonholonomic mechanics

Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems

Contact Info

Product

Resources

About