Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 421above, as well as [Marsden and Ratiu (1999); Ivancevic and Snoswell (2001);Ivancevic (2002); Ivancevic and Pearce (2001b); Ivancevic (2005)]). Recallthat the Riemannian metric g = on the configuration manifold M isa positive–definite quadratic form g : T M → R, given in local coordinatesq i ∈ U (U open in M) asg ij ↦→ g ij (q, m) dq i dq j ,∂x r ∂x sg ij (q, m) = m µ δ rs∂q i ∂q jwhereis the covariant material metric tensor g, defining a relation between internaland external coordinates and including n segmental masses m µ .The quantities x r are external coordinates (r, s = 1, . . . , 6n) and i, j =1, . . . , N ≡ 6n − h, where h denotes the number of holonomic constraints.The Lagrangian of the system is a quadratic form L : T M → R dependenton velocity v and such that L(v) = 1 2< v, v >. It is given byL(v) = 1 2 g ij(q, m) v i v jin local coordinates q i , v i = ˙q i ∈ U v (U v open in T M). The Hamiltonianof the system is a quadratic form H : T ∗ M → R dependent on momentump and such that H(p) = 1 2< p, p >. It is given byH(p) = 1 2 gij (q, m) p i p jin local canonical coordinates q i , p i ∈ U p (U p open in T ∗ M). The inverse(contravariant) metric tensor g −1 , is defined asg ij (q, m) = m µ δ rs∂q i∂x r ∂q j∂x s .For any smooth function L on T M, the fibre derivative, or Legendretransformation, is a diffeomorphism FL : T M → T ∗ M, F(w) · v = , from the momentum phase–space manifold to the velocity phase–space manifold associated with the metric g = . In local coordinatesq i , v i = ˙q i ∈ U v (U v open in T M), FL is given by (q i , v i ) ↦→ (q i , p i ).Recall that on the momentum phase–space manifold T ∗ M exists:(i) A unique canonical 1−form θ H with the property that, for any 1−formβ on the configuration manifold M, we have β ∗ θ H = β. In local canonicalcoordinates q i , p i ∈ U p (U p open in T ∗ M) it is given by θ H = p i dq i .(ii) A unique nondegenerate Hamiltonian symplectic 2−form ω H , which is