Dirac structures and geometry of nonholonomic constraints

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Constrained Lagrangian system according to Y & M Definition A partial vector field on T ∗ Q is a map such that X : TQ ⊕Q T ∗ Q −→ TT ∗ Q X (v, p) ∈ TpT ∗ Q. KG (KMMF) Seminarium Geometryczne 13 X 2010 11 / 26
Constrained Lagrangian system according to Y & M A Lagrangian system with nonholonomic constraints ∆Q can be described by a triple (L, ∆, X ) where L is a Lagrangian function, possibly not hyperregular, ∆ is the Dirac structure associated to the constraint distribution ∆Q, X is a partial vector field defined in points and such that (v, p) ∈ TQ ⊕ T ∗ Q, such that v ∈ ∆Q, p = FL(v) X (v, p) + DL(v) ∈ ∆ KG (KMMF) Seminarium Geometryczne 13 X 2010 12 / 26
Page 1 and 2: Dirac structures and geometry of no
Page 3 and 4: Introduction References Hiroaki Yos
Page 5 and 6: Contents Dirac structure Dirac stru
Page 11 and 12: Dirac structure Definition A Dirac
Page 13 and 14: Almost Dirac structure in mechanics
Page 19 and 20: Almost Dirac structures in mechanic
Page 25 and 26: Constrained Lagrangian system accor
Page 27: Constrained Lagrangian system accor
Page 35 and 36: My point of view Lagrangian dynamic
Page 37 and 38: My point of view Back We treat ∆
Page 39 and 40: My point of view Back We treat ∆
Page 41 and 42: My point of view How to get ˜ ∆
Page 47 and 48: My point of view The constrained ca
Page 49 and 50: My point of view The constrained ca
Page 51 and 52: My point of view ˜∆ KG (KMMF) Se
Page 53 and 54: My point of view Conclusion The rel
Page 55 and 56: Example Equations for a curve t ↦
Page 57 and 58: Example Initial momentum along the

Dirac structures and geometry of nonholonomic constraints

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