Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 381The vector space R 3 is a Lie algebra with respect to the bracket operationgiven by the usual cross product. The space R 3 is paired with itselfvia the usual dot product. So if F : R 3 → R, then δF /δp = ∇F (p) andthe (–) Lie–Poisson bracket {F, G} − (p) is given via (3.178) by the tripleproduct{F, G} − (p) = −p · (∇F (p) × ∇G(p)).Euler’s vector equation (3.190) represents a generalized Hamiltoniansystem in R 3 relative to the Hamiltonian function H(p) and the (–) Lie–Poisson bracket {F, G} − (p). Thus the Poisson manifold (R 3 , {F, G} − (p)) isdefined and the abstract Poisson equation is equivalent to Euler’s equation(3.190) for a body segment and associated joint.Solitary Model of Muscular ContractionRecall that the so–called sliding filament theory of muscular contractionwas developed in 1950s by Nobel Laureate A. Huxley [Huxley andNiedergerke (1954); Huxley (1957)]. At a deeper level, the basis of themolecular model of muscular contraction is represented by oscillationsof Amid I peptide groups with associated dipole electric momentum insidea spiral structure of myosin filament molecules (see [Davydov (1981);Davydov (1991)]).There is a simultaneous resonant interaction and strain interaction generatinga collective interaction directed along the axis of the spiral. Theresonance excitation jumping from one peptide group to another can berepresented as an exciton, the local molecule strain caused by the staticeffect of excitation as a phonon and the resultant collective interaction asa soliton.The simplest model of Davydov’s solitary particle–waves is given by thenonlinear Schrödinger equation [Ivancevic and Pearce (2001a)]i∂ t ψ = −∂ x 2ψ + 2χ|ψ| 2 ψ, (3.191)for -∞ < x < +∞. Here ψ(x, t) is a smooth complex–valued wave functionwith initial condition ψ(x, t)| t=0 = ψ(x) and χ is a nonlinear parameter. Inthe linear limit (χ = 0) (3.191) becomes the ordinary Schrödinger equationfor the wave function of the free 1D particle with mass m = 1/2.We may define the infinite–dimensional phase–space manifold P ={(ψ, ¯ψ) ∈ S(R, C)}, where S(R, C) is the Schwartz space of rapidly–decreasing complex–valued functions defined on R). We define also the