Improving E. Cartan considerations on the invariance of nonholonomic mechanics

Oliva, Waldyr M.; Terra, Gláucio

doi:10.3934/jgm.2019022

Cited by 1 publication

(7 citation statements)

References 7 publications

(26 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Since the distribution spanned by 4 , 5 , 6 is not orthogonal to when ≠ 0, the nh connections are different. Terra and Oliva observed in [76] that one gets the same nonholonomic paths.…”

Section: In Order To Show Thatmentioning

confidence: 91%

“…Section 3 gives the initial -structure for the equivalence method. Section 4 builds on recent work by Terra and Oliva [76]. They consider a weaker notion of equivalence, the preservation of the acceleration instead of the nonholonomic connection .…”

Section: Chaplygin and Hamiltonizationmentioning

confidence: 99%

“…Cartan's note was revisited in [63]. As pointed out by Terra and Oliva [76], there is an inaccuracy. It was asserted that Cartan's equivalence criterion (see section 3.2) provided necessary and sufficient conditions for a change of metric in to keep the straightest paths.…”

Section: Cartan's Equivalence For Nh Connectionsmentioning

confidence: 99%

“…We present situations where the notions coincide in the next section 4. The counter example by Terra an Oliva [76] is reviewed in section 4.6.…”

Section: G-structures In Nonholonomic Geometrymentioning

confidence: 99%

“…Our goal is to discuss the equivalence problem of Cartan's nonholonomic connection under changes of metric. We pursue a recent observation by Terra and Oliva [76] that there is more freedom if one is happy by keeping the same straightest paths = 0. We show nonetheless that in many cases the notions coincide.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Cartan meets Chaplygin

Ehlers

Koiller

2019

Theor appl mech (Belgr)

View full text Add to dashboard Cite

In a note at the 1928 International Congress of Mathematicians Cartan outlined how his "method of equivalence" can provide the invariants of nonholonomic systems on a manifold with kinetic lagrangians [29]. Cartan indicated which changes of the metric outside the constraint distribution ⊂ preserve the nonholonomic connection = Proj ∇ , , ∈ , where ∇ is the Levi-Civita connection on and Proj is the orthogonal projection over. Here we discuss this equivalence problem of nonholonomic connections for Chaplygin systems [30, 31, 62]. We also discuss an example-a mathematical gem!-found by Oliva and Terra [76]. It implies that there is more freedom (thus more opportunities) using a weaker equivalence, just to preserve the straightest paths: = 0. However, finding examples that are weakly but not strongly equivalent leads to an over-determined system of equations indicating that such systems should be rare. We show that the two notions coincide in the following cases: i) Rank two distributions. This implies for instance that in Cartan's example of a sphere rolling on a plane without slipping or twisting, a (2,3,5) distribution, the two notions of equivalence coincide; ii) For a rank 3 or higher distribution, the corank of D in D+[D,D] must be at least 3 in order to find examples where the two notions of equivalence do not coincide. This rules out the possibility of finding examples on (3,5) distributions such as Chaplygin's marble sphere. Therefore the beautiful (3,6) example by Oliva and Terra is minimal.

show abstract

“…Since the distribution spanned by 4 , 5 , 6 is not orthogonal to when ≠ 0, the nh connections are different. Terra and Oliva observed in [76] that one gets the same nonholonomic paths.…”

Section: In Order To Show Thatmentioning

confidence: 91%

Section: Chaplygin and Hamiltonizationmentioning

confidence: 99%