Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1043force in the system–environment coupling arising from the imaginary partof the action (see [Srivastava et al. (1995)]).When in (6.72) we useL(ẋ, ẏ, x, y) = mẋẏ − VF rety = γẏ and F advx = −γẋ we get,(x + 1 )2 y + V(x − 1 )2 y + γ (xẏ − yẋ). (6.74)2By usingV(x ± 1 )2 y = 1 2 κ(x ± 1 2 y)2in (6.74), the DHO equation (6.63) and its complementary equation for they coordinatemÿ − γẏ + κy = 0. (6.75)are derived. The y−oscillator is the time–reversed image of the x−oscillator(6.63). From the manifolds of solutions to equations (6.63) and (6.75), wecould choose those for which the y coordinate is constrained to be zero,they simplify tomẍ + γẋ + κx = 0, y = 0.Thus we get the classical damped oscillator equation from a Lagrangiantheory at the expense of introducing an ‘extra’ coordinate y, later constrainedto vanish. Note that the constraint y(t) = 0 is not in violation ofthe equations of motion since it is a true solution to (6.63) and (6.75).6.3.9 Application: Cerebellum as a Neural Path–IntegralRecall that human motion is naturally driven by synergistic action of morethan 600 skeletal muscles. While the muscles generate driving torques inthe moving joints, subcortical neural system performs both local and global(loco)motion control: first reflexly controlling contractions of individualmuscles, and then orchestrating all the muscles into synergetic actions inorder to produce efficient movements. While the local reflex control ofindividual muscles is performed on the spinal control level, the global integrationof all the muscles into coordinated movements is performed withinthe cerebellum.