Ivancevic_Applied-Diff-Geom

422 Applied Differential Geometry: A Modern Introductionclosed (dω H = 0) and exact (ω H = dθ H = dp i ∧dq i ). Each body segmenthas, in the general SO(3) case, a sub–phase–space manifold T ∗ SO(3) withω (sub)H= dp φ ∧ dφ + dp ψ ∧ dψ + dp θ ∧ dθ.Analogously, on the velocity phase–space manifold T M exists:(i) A unique 1−form θ L , defined by the pull–back θ L = (FL) ∗ θ H of θ Hby FL. In local coordinates q i , v i = ˙q i ∈ U v (U v open in T M) it is givenby θ L = L v idq i , where L v i ≡ ∂L/∂v i .(ii) A unique nondegenerate Lagrangian symplectic 2−form ω L , defined bythe pull–back ω L = (FL) ∗ ω H of ω H by FL, which is closed (dω L = 0)and exact (ω L = dθ L = dL v i ∧ dq i ).Both T ∗ M and T M are orientable manifolds, admitting the standardvolumes given respectively byΩ ωH , =N(N+1)(−1) 2N!ω N H, and Ω ωL =N(N+1)(−1) 2N!ω N L ,in local coordinates q i , p i ∈ U p (U p open in T ∗ M), resp. q i , v i = ˙q i ∈ U v(U v open in T M). They are given byΩ H = dq 1 ∧ · · · ∧ dq N ∧ dp 1 ∧ · · · ∧ dp N , andΩ L = dq 1 ∧ · · · ∧ dq N ∧ dv 1 ∧ · · · ∧ dv N .On the velocity phase–space manifold T M we can also define the actionA : T M → R by A(v) = FL(v) · v and the energy E = A − L. In localcoordinates q i , v i = ˙q i ∈ U v (U v open in T M) we have A = v i L v i, soE = v i L v i − L. The Lagrangian vector–field X L on T M is determinedby the condition i XL ω L = dE. Classically, it is given by the second–orderLagrangian equationsddt L v i = L qi. (3.233)The Hamiltonian vector–field X H is defined on the momentum phase–space manifold T ∗ M by the condition i XH ω = dH. ( The condition ) may be0 Iexpressed equivalently as X H = J∇H, where J = .−I 0In local canonical coordinates q i , p i ∈ U p (U p open in T ∗ M) the vector–field X H is classically given by the first–order Hamiltonian canonical equations˙q i = ∂ pi H,p˙i = − ∂ q iH. (3.234)