Differential geometry and complex analysis: a volume by I. Chavel, H.M. Farkas

By I. Chavel, H.M. Farkas

Chavel I., Farkas H.M. (eds.) Differential geometry and intricate research (Springer, 1985)(ISBN 354013543X)(236s)

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21 The vector g ij ∂i S∂j is orthogonal to the surface of constant S . In fact it is just the tangent vector of unit length to the geodesic connecting the point to the origin. Thus g ij (x)∂i S∂j S = 1. This first order partial differential equation is called the eikonal equation. It is just another way of describing geodesics of a manifold: the geodesics are its characteristic curves. It is analogous to the Hamilton-Jacobi equation in Mechanics. Often it is easier to solve this equation than the ordinary differential equation above.

It can depend on position. 8 . 3 Given a curve r : [a, b] → R3 in space, its optical length is defined to be ab n(r(t))|˙r(t)|dt . This is the length of the curve with respect to the optical metric ds2o = n(x)2 [dx2 + dy 2 + dz 2 ] . This is a Riemann metric on R3 which can differ from the Euclidean metric by the conformal factor n(x) . The angles between vectors with respect to the optical metric are the same as in Euclidean geometry, but the lengths of vectors (and hence curves) are different.

G. 7 The set of all possible instantaneous states of a classical mechanical system is a symplectic manifold, called its phase space. An observable is a real valued function on the phase space. From an observable f we can construct its canonical vector field Xf . 8 Time evolution is given by the integral curves of the canonical vector field of a particular function, called the hamiltonian. Thus a classical mechanical system is defined by a triple (M, ω, H) : a differential manifold M , a symplectic form ω on it and a smooth function H : M → R .

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