Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 387tor Can defines the first–order canonical Hamiltonian formalism on thecotangent bundle T ∗ M (i.e., the momentum phase–space manifold). Asthese two formalisms are related by the isomorphic functor Dual, they areequivalent. In this section we shall follow the Lagrangian functor Lie, usingthe powerful formalism of exterior differential systems and integral variationalprinciples [Griffiths (1983); Choquet-Bruhat and DeWitt-Morete(1982)]. For the parallel, Hamiltonian treatment along the functor Can,more suitable for chaos theory and stochastic generalizations, see [Ivancevicand Snoswell (2001); Ivancevic (2002)].Exterior Lagrangian DynamicsLet Ω p (M) = ∑ ω I dx I denote the space of differential p−forms on M.That is, if multi–index I ⊂ {1, . . . , n} is a subset of p elements then wehave a p−form dx I = dx i 1 ∧ dx i 2 ∧ · · · ∧ dx i p on M. We define the exteriorderivative on M as dω = ∑ ∂ω I∂x pdx p ∧ dx I (compare with (5.8) above).Now, from exterior differential systems point of view (see subsection3.6.2 above as well as [Griffiths (1983)]), human–like motion representsan n DOF neuro–musculo–skeletal system Ξ, evolving in time on its nDconfiguration manifold M, (with local coordinates x i , i = 1, ..., n) as wellas on its tangent bundle T M (with local coordinates (x i ; ẋ i )).For the system Ξ we will consider a well–posed variational problem(I, ω; ϕ), on an associated (2n+1)−-D jet space X = J 1 (R, M) ∼ = R×T M,with local canonical variables (t; x i ; ẋ i ).Here, (I, ω) is called a Pfaffian exterior differential system on X (see[Griffiths (1983)]), given locally as{ θ i = dx i − ẋ i ω = 0, (3.195)ω ≡ dt ≠ 0with the structure equationsdθ i = −dẋ i ∧ ω.Integral manifolds N ∈ J 1 (R, M) of the Pfaffian system (I, ω) are locallyone–jets t → (t, x(t), ẋ(t)) of curves x = x(t) : R → M.ϕ is a 1−formϕ = L ω, (3.196)where L = L(t, x, ẋ) is the system’s Lagrangian function defined on X, havingboth coordinate and velocity partial derivatives, respectively denoted