Symmetries and currents in nonholonomic mechanics

ČECH, Michal and Jana MUSILOVÁ. Symmetries and currents in nonholonomic mechanics. Communications in Mathematics. Ostrava, CR: Ostravska univerzita v Ostrave, 2014, 22/2014, No 2, p. 159-184. ISSN 1804-1388.

Basic information

• Original name: Symmetries and currents in nonholonomic mechanics
• Name in Czech: Symetrie a toky v neholonomni­ mechanice.
• Authors: ČECH, Michal (203 Czech Republic, belonging to the institution) and Jana MUSILOVÁ (203 Czech Republic, guarantor, belonging to the institution).
• Edition: Communications in Mathematics, Ostrava, CR, Ostravska univerzita v Ostrave, 2014, 1804-1388.

Other information

• Original language: English
• Type of outcome: Article in a journal
• Field of Study: 10301 Atomic, molecular and chemical physics
• Country of publisher: Czech Republic
• Confidentiality degree: is not subject to a state or trade secret
• RIV identification code: RIV/00216224:14310/14:00074352
• Organization unit: Faculty of Science
• Keywords (in Czech): neholonomni­ mechanicke systemy; neholonomni vazebni­ podvarieta; kanonicka distribuce; redukovane pohybove rovnice; symetrie neholonomnich systemu; zakony zachovani­; Chaplyginovy sane›
• Keywords in English: nonholonomic mechanical systems; nonholonomic constraint submanifold; canonical
• Tags: AKR, rivok
• Tags: International impact, Reviewed

Abstract

In this paper we derive general equations for constraint Noether- -type symmetries of a rst order non-holonomic mechanical system and the corresponding currents, i.e. functions constant along trajectories of the nonholonomic system. The approach is based on a consistent and very ef- fective geometrical theory of nonholonomic constrained systems on bred manifolds and their jet prolongations, rst presented and developed by Olga Rossi. As a representative example of application of the geometrical theory and the equations of symmetries and conservation laws derived within this framework we present the Chaplygin sleigh. It is a mechanical system sub- ject to one linear nonholonomic constraint enforcing the plane motion. We describe the trajectories of the Chaplygin sleigh and show that the usual kinetic energy conservation law holds along them, the time translation gen- erator being the corresponding constraint symmetry and simultaneously the symmetry of nonholonomic equations of motion. Moreover, the expressions for two other currents are obtained. The corresponding constraint symme- tries are not symmetries of nonholonomic equations of motion. The physical interpretation of results is emphasized.

Abstract (in Czech)

Jsou odvozeny obecne rovnice pro vazane noetherovske symetrie neholonomniho mechanickeho systemu a odpovidaji­ci­ toky (veliciny zachovavaji­ci­ se podel trajektorii­). Pristup je zalozen na efektivni geometricke teorii koncipovane a rozvijene Olgou Rossi. Jako reprezentativni pri­klad je uveden neholonomni­ system v jednou linearni neholonomni­ vazbou konaji­ci­ rovinny pohyb, tzv. Chaplyginovy sane. Jsou reseny redukovane pohybove rovnice a prezentovany graficke vystupy. Jednou z vazanych symetrii je operator casove translace, jemuz odpovi­da zakon zachovani­ mechanicke energie. Jsou nalezeny dalsi­ dve vazane symetrie a jim odpovidaji­ci toky. Jsou zi­skany vyrazy pro Chetaevovy vazebni­ si­ly. Duraz je kladen na fyzikalni interpretaci vysledku.

Links

• GA14-02476S, research and development project: Name: Variace, geometrie a fyzika

Investor: Czech Science Foundation