Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 277which is a linear map. It is also called the Laplace–Beltrami operator, sinceBeltrami first considered this operator on Riemannian manifolds.Riemannian metric has the following mechanical interpretation. Let Mbe a closed Riemannian manifold with the mechanical metric g = g ij v i v j ≡〈v, v〉, with v i = ẋ i . Consider the Lagrangian functionL : T M → R, (x, v) ↦→ 1 〈v, v〉 − U(x) (3.130)2where U(x) is a smooth function on M called the potential. On a fixedlevel of energy E, bigger than the maximum of U, the Lagrangian flowgenerated by (3.130) is conjugate to the geodesic flow with metric ḡ =2(e − U(x))〈v, v〉. Moreover, the reduced action of the Lagrangian is thedistance for g = 〈v, v〉 [Arnold (1989); Abraham et al. (1988)]. Both ofthese statements are known as the Maupertius action principle.3.10.1.2 Geodesics on MFor a C k , k ≥ 2 curve γ : I → M, we define its length on I as∫ ∫√L (γ, I) = | ˙γ| dt = g ( ˙γ, ˙γ)dt.IThis length is independent of our parametrization of the curve γ. Thus thecurve γ can be reparameterized, in such a way that it has unit velocity.The distance between two points m 1 and m 2 on M, d (m 1 , m 2 ) , can now bedefined as the infimum of the lengths of all curves from m 1 to m 2 , i.e.,IL (γ, I) → min .This means that the distance measures the shortest way one can travel fromm 1 to m 2 .If we take a variation V (s, t) : (−ε, ε) × [0, l] → M of a smooth curveγ (t) = V (0, t) parameterized by arc–length L and of length l, then thefirst derivative of the arc–length functionL(s) =dL(0)ds∫ l0| ˙V | dt, is given by∫ l≡ ˙L(0) = g ( ˙γ, X)| l 0 − g (γ, X) dt, (3.131)where X (t) = ∂V∂s(0, t) is the so–called variation vector–field. Equation(3.131) is called the first variation formula. Given any vector–field X alongγ, one can produce a variation whose variational field is X. If the variation0