Dirac structures and geometry of nonholonomic constraints
Dirac structures and geometry of nonholonomic constraints Dirac structures and geometry of nonholonomic constraints
My point of view Back We treat ∆ as a relation T ∗ T ∗ Q −− ⊲ TT ∗ Q We can take a ”short-cut” identifying T ∗ T ∗ Q with T ∗ TQ Local expressions: TT∗Q❊ T ✡ TπQ ❊ ✡ ❊ τT∗Q ✡ ✡ ✡ ✡ ∗ ˜∆ ✤ TQ dL ✡ ❊ ❊ ζ ✡ π ❊ TQ ✡ ∆Q ✡ ∆Q ✡ ✡ T∗Q T∗Q T ∗ TO O × V × V ∗ × V ∗ ∋ (q, ˙q, c, d) TT ∗ O O × V ∗ × V × V ∗ ∋ (q, p, ˙q, ˙p) ˜∆ : ˙q ∈ ∆Q(q), p = d ˙p − c ∈ ∆ ◦ Q (q) KG (KMMF) Seminarium Geometryczne 13 X 2010 16 / 26
My point of view How to get ˜ ∆ not going to the ”Hamiltonian” side of the diagram? Many of the ideas od Analytical Mechanics come from statics, In statics constraints are described by admissible configurations and admissible virtual displacements (WMT), We observed in ”Variational ... in general algebroids” that different set of admissible virtual displacements produce different E-L equations (vaconomic, nonholonomic) for the same set of admissible configurations. How to put virtual displacements into the game? KG (KMMF) Seminarium Geometryczne 13 X 2010 17 / 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 7 and 8: Contents Dirac structure Dirac stru
- Page 9 and 10: Contents Dirac structure Dirac stru
- Page 11 and 12: Dirac structure Definition A Dirac
- Page 13 and 14: Almost Dirac structure in mechanics
- Page 15 and 16: Almost Dirac structure in mechanics
- Page 17 and 18: Almost Dirac structure in mechanics
- Page 19 and 20: Almost Dirac structures in mechanic
- Page 21 and 22: Almost Dirac structures in mechanic
- Page 23 and 24: Almost Dirac structures in mechanic
- Page 25 and 26: Constrained Lagrangian system accor
- Page 27 and 28: Constrained Lagrangian system accor
- Page 29 and 30: Constrained Lagrangian system accor
- Page 31 and 32: Constrained Lagrangian system accor
- Page 33 and 34: 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: My point of view Back We treat ∆
- Page 43 and 44: My point of view How to get ˜ ∆
- Page 45 and 46: 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
My point <strong>of</strong> view<br />
How to get ˜ ∆ not going to the ”Hamiltonian” side <strong>of</strong> the diagram?<br />
Many <strong>of</strong> the ideas od Analytical Mechanics come from statics,<br />
In statics <strong>constraints</strong> are described by admissible configurations <strong>and</strong><br />
admissible virtual displacements (WMT),<br />
We observed in ”Variational ... in general algebroids” that different<br />
set <strong>of</strong> admissible virtual displacements produce different E-L<br />
equations (vaconomic, <strong>nonholonomic</strong>) for the same set <strong>of</strong> admissible<br />
configurations.<br />
How to put virtual displacements into the game?<br />
KG (KMMF) Seminarium Geometryczne 13 X 2010 17 / 26
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