By Yuri Gliklikh

The first variation of this publication entitled research on Riemannian Manifolds and a few difficulties of Mathematical Physics was once released by means of Voronezh Univer­ sity Press in 1989. For its English variation, the e-book has been considerably revised and elevated. specifically, new fabric has been additional to Sections 19 and 20. i'm thankful to Viktor L. Ginzburg for his exertions at the transla­ tion and for writing Appendix F, and to Tomasz Zastawniak for his quite a few feedback. My specified thank you visit the referee for his invaluable feedback at the concept of stochastic techniques. eventually, i want to recognize the help of the AMS fSU relief Fund and the overseas technological know-how beginning (Grant NZBOOO), which made attainable my paintings on a number of the new effects integrated within the English variation of the publication. Voronezh, Russia Yuri Gliklikh September, 1995 Preface to the Russian variation the current publication is seemingly the 1st in monographic literature during which a standard therapy is given to 3 parts of worldwide research formerly consid­ ered fairly far away from one another, particularly, differential geometry and classical mechanics, stochastic differential geometry and statistical and quantum me­ chanics, and infinite-dimensional differential geometry of teams of diffeomor­ phisms and hydrodynamics. The unification of those themes below the canopy of 1 booklet seems to be, although, rather typical, because the exposition is predicated on a geometrically invariant type of the Newton equation and its analogs taken as a basic legislation of motion.

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The operator G is called the inertia operator (or the inertia tensor). In particular, p = G(rin). In terms of the inertia operator, the kinetic energy of the system is given by the equation 1C (X) = (G(X))(X)/2. , the standard inner product in 1R3), which is not related to the metric (, ) 4. Geometric Mechanics: Introduction and Review of Standard Examples 19 giving rise to K. In this case, we may identify vectors and covectors by means of (,) and view G as an operator from TM to itself. Then (X, Y) = (LX, Y) and G is self-adjoint.

Accessible Points of Mechanical Systems 9. Examples of Points that Cannot Be Connected by a Trajectory Example 1 ([621 and [66]). Consider the mechanical system on the unit sphere S2 in JR3 with the force field a(f) = (-y, x, 0), where r = (x, y, z) E S2. z2 2 2 is the kinetic energy. , [119]) to the holonomic constraint F(r) = x2 + y2 + z2. Denote the North and South Pole of the sphere by N = (0, 0, 1) and S = (0, 0, -1), respectively. Let r(t) = (x(t), y(t), z(t)) be the trajectory of the system such that r(to) = S for some to and T(to) = V 0.

1. , [17]). Let m(t), t E I, be a Cl-curve in M. The development 6(m(t)) of m(t) is a curve in Tm(o)M, which can be constructed as follows. ), thus obtaining a curve in Tm(O)M. Then 6(m(t)) is set to be the antiderivative of this curve. In other words, (m(t)) = f T(Th(t)) drr . The operator 6: C m10 (I, M) -+ C' (I, TmOM) is invertible: 6-1(u(t)) = Sit(t) for u E C' (I, Tm0 M). For what follows it is important to emphasize that 14 Chapter 1. 1. To see this, let and replace the basis b° by b1 in Tm0M.

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