Gauge conservation laws and the momentum equation in nonholonomic mechanics

Fassoò, Francesco; Giacobbe, Andrea; Sansonetto, Nicola

doi:10.1016/s0034-4877(09)00005-6

Cited by 22 publications

(80 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As was shown in [21], this gauge mechanism is closely related (equivalent, if e.g. the action is locally free) to the so-called "momentum equation" and "nonholonomic momentum map" [8, 7, 12-16, 34, 32].…”

Section: Theorem 3 ([22])mentioning

confidence: 94%

See 1 more Smart Citation

An Elemental Overview of the Nonholonomic Noether Theorem

Fassò

Sansonetto

2009

Int. J. Geom. Methods Mod. Phys.

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Theorem 3 ([22])mentioning

confidence: 94%

“…(Here, G is the distribution on Q whose fibers Note that, based on Theorem 2, this "gauge" method obviously extends to sections of R • , see [21]. Whether the gauge method should be considered as a fundamental mechanism which links symmetries and conservation laws is presently unclear.…”

Section: Theorem 3 ([22])mentioning

confidence: 99%

An Elemental Overview of the Nonholonomic Noether Theorem

Fassò

Sansonetto

2009

Int. J. Geom. Methods Mod. Phys.

Self Cite

View full text Add to dashboard Cite

show abstract

“…This contributes to the recent efforts to understand the mechanisms responsible for the existence of first integrals that are linear in velocities in nonholonomic mechanics (see e.g. [24,11,12,1]).…”

Section: Introductionmentioning

confidence: 91%

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

“…We will follow the Lagrangian description in this paper, although the Hamiltonian description of this problem is also of interest and will be considered in later work. Our goal is to prove that integrals given by (18) exists if and only if A 1 = 0. Moreover, we shall prove that there are exactly two independent constants of motion of that type, which is sufficient for integrability.…”

Section: Extra Constants Of Motion and Conditions For Integrabilitymentioning

confidence: 99%

Integrability and Chaos in Figure Skating

Gzenda

Putkaradze

2019

J Nonlinear Sci

View full text Add to dashboard Cite

We derive and analyze a three dimensional model of a figure skater. We model the skater as a three-dimensional body moving in space subject to a non-holonomic constraint enforcing movement along the skate's direction and holonomic constraints of continuous contact with ice and pitch constancy of the skate. For a static (non-articulated) skater, we show that the system is integrable if and only if the projection of the center of mass on skate's direction coincides with the contact point with ice and some mild (and realistic) assumptions on the directions of inertia's axes. The integrability is proved by showing the existence of two new constants of motion linear in momenta, providing a new and highly nontrivial example of an integrable non-holonomic mechanical system. We also consider the case when the projection of the center of mass on skate's direction does not coincide with the contact point and show that this non-integrable case exhibits apparent chaotic behavior, by studying the divergence of nearby trajectories. We also demonstrate the intricate behavior during the transition from the integrable to chaotic case. Our model shows many features of reallife skating, especially figure skating, and we conjecture that real-life skaters may intuitively use the discovered mechanical properties of the system for the control of the performance on ice.

show abstract

Gauge conservation laws and the momentum equation in nonholonomic mechanics

Cited by 22 publications

References 36 publications

An Elemental Overview of the Nonholonomic Noether Theorem

An Elemental Overview of the Nonholonomic Noether Theorem

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

Integrability and Chaos in Figure Skating

Contact Info

Product

Resources

About