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 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
- 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: My point of view Back We treat ∆
- Page 41 and 42: My point of view How to get ˜ ∆
- 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 />
Back<br />
We treat ∆ as a relation T ∗ T ∗ Q −− ⊲ TT ∗ Q<br />
We can take a ”short-cut” identifying T ∗ T ∗ Q with T ∗ TQ<br />
Local expressions:<br />
TT∗Q❊ T<br />
✡ TπQ ❊<br />
✡ ❊<br />
τT∗Q ✡<br />
✡<br />
✡<br />
✡<br />
∗ ˜∆ <br />
✤<br />
TQ <br />
dL<br />
✡<br />
❊<br />
❊<br />
ζ ✡ π ❊ TQ <br />
✡<br />
∆Q ✡ ∆Q<br />
✡<br />
✡<br />
T∗Q T∗Q T ∗ TO O × V × V ∗ × V ∗ ∋ (q, ˙q, c, d)<br />
TT ∗ O O × V ∗ × V × V ∗ ∋ (q, p, ˙q, ˙p)<br />
˜∆ : ˙q ∈ ∆Q(q), p = d ˙p − c ∈ ∆ ◦ Q (q)<br />
KG (KMMF) Seminarium Geometryczne 13 X 2010 16 / 26
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