Dirac structures and geometry of nonholonomic constraints
Dirac structures and geometry of nonholonomic constraints Dirac structures and geometry of nonholonomic constraints
Example Lagrangian system with constraints: a sleigh on a horizontal plane Q = R 2 × S 1 ∋ (x, y, ϕ), ∆Q(x, y, ϕ) = {( ˙x, ˙y, ˙ϕ) : ˙x sin(ϕ) = ˙y cos(ϕ)}, ∆Q = 〈 ∂ ∂ , cos(ϕ) ∂ϕ ∂x + sin(ϕ) ∂ ∂y 〉 L(x, y, ϕ, ˙x, ˙y, ˙ϕ) = m 2 ( ˙x 2 + ˙y 2 ) + I 2 ˙ϕ2 KG (KMMF) Seminarium Geometryczne 13 X 2010 22 / 26
Example Equations for a curve t ↦−→ (x(t), y(t), ϕ(t), px(t), py (t), π(t)) ˙x = px/m, ˙px = µ(t) sin ϕ ˙y = py /m, ˙py = −µ(t) cos ϕ ˙ϕ = π/m, ˙π = 0 ˙x sin ϕ = ˙y cos ϕ Solution (momentum) π(t) = π0 px(t) = p0 cos( π0 t + ϕ0) I py (t) = p0 sin( π0 t + ϕ0) I KG (KMMF) Seminarium Geometryczne 13 X 2010 23 / 26
- 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 and 40: 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: My point of view Conclusion The rel
- Page 57 and 58: Example Initial momentum along the
Example<br />
Lagrangian system with <strong>constraints</strong>: a sleigh on a horizontal plane<br />
Q = R 2 × S 1 ∋ (x, y, ϕ),<br />
∆Q(x, y, ϕ) = {( ˙x, ˙y, ˙ϕ) : ˙x sin(ϕ) = ˙y cos(ϕ)},<br />
∆Q = 〈 ∂ ∂<br />
, cos(ϕ)<br />
∂ϕ ∂x<br />
+ sin(ϕ) ∂<br />
∂y 〉<br />
L(x, y, ϕ, ˙x, ˙y, ˙ϕ) = m<br />
2 ( ˙x 2 + ˙y 2 ) + I<br />
2 ˙ϕ2<br />
KG (KMMF) Seminarium Geometryczne 13 X 2010 22 / 26
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