Ivancevic_Applied-Diff-Geom

Introduction 25tons. In this formulation, particles travel every possible path between theinitial and final states; the probability of a specific final state is obtained bysumming over all possible trajectories leading to it. In the classical regime,the path integral formulation cleanly reproduces the Hamilton’s principle,as well as the Fermat’s principle in optics.1.1.5.4 Finsler manifoldsFinsler manifolds represent generalization of Riemannian manifolds. AFinsler manifold allows the definition of distance, but not of angle; it isan analytic manifold in which each tangent space is equipped with a norm‖.‖ in a manner which varies smoothly from point to point. This norm canbe extended to a metric, defining the length of a curve; but it cannot ingeneral be used to define an inner product. Any Riemannian manifold (butnot a pseudo–Riemannian manifold) is a Finsler manifold. 311.1.6 Symplectic Manifolds: Phase–Spaces for HamiltonianMechanicsA symplectic manifold is a smooth manifold M equipped with a closed, nondegenerate,2–form ω called the symplectic volume form, or Liouville measure.This condition forces symplectic manifolds to be even–dimensional.Cotangent bundles, which arise as phase–spaces in Hamiltonian mechanics,are the motivating example, but many compact manifolds also have symplecticstructure. All surfaces have a symplectic structure, since a symplecticstructure is simply a volume form. The study of symplectic manifoldsis called symplectic geometry/topology.Symplectic manifolds arise naturally in abstract formulations of classicalmechanics as the cotangent bundles of configuration manifolds: the set ofall possible configurations of a system is modelled as a manifold M, andthis manifold’s cotangent bundle T ∗ M describes the phase–space of the31 Formally, a Finsler manifold is a differentiable manifold M with a Banach normdefined over each tangent space such that the Banach norm as a function of position issmooth, usually it is assumed to satisfy the following regularity condition:For each point x of M, and for every nonzero vector X in the tangent space T × M,the second derivative of the function L : T × M → R given by L(w) = 1 2 ‖w‖2 at X ispositive definite.The length of a smooth curve γ in a Finsler manifold M is given by R ‚ ‚ dγ ‚‚ ‚ dt (t) dt.Length is invariant under reparametrization. With the above regularity condition,geodesics are locally length–minimizing curves with constant speed, or equivalentlycurves in whose energy function, R ‚ ‚ ‚dγdt (t) ‚ ‚‚ 2dt, is extremal under functional derivatives.