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General vortex theory. (Obshchaya teoriya vikhreĭ.) (Russian) Zbl 1054.37514
Izhevsk: Nauchno-Izdatel’skij Tsentr “Regulyarnaya i Khaoticheskaya Dinamika” (ISBN 5-7029-0299-8/pbk). 238 p. (1998).
This is a very interesting book that investigates the connections between hydrodynamics on the one side and classical mechanics, geometrical optics and symplectic geometry on the other side. At the core of these connections lies a generalization of the well-known Lamb equation in hydrodynamics, which describes the motion of a barotropic perfect fluid under potential forces.
The book has four chapters and two appendices.
Chapter I presents a detailed and well-written introduction to the parts of hydrodynamics, classical mechanics and geometrical optics where the Lamb equation and its generalization appear. For example, the Hamilton-Jacobi equation, which appears in a natural way in geometrical optics and classical mechanics, is a particular case of the generalized Lamb equation.
In Chapter II, the main properties of the generalized Lamb equation are given. The generalized Lamb equation (hereafter called simply “the Lamb equation”) is written as follows. Let $$M^n$$ be a smooth manifold and let $$u=u(x,t)\in T^{*}_xM^n$$ be a smooth co-vector field on $$M^n$$ that depends smoothly on the time $$t$$. Consider a smooth vector field $$v=v(x,t)$$ on the manifold $$M^n$$. The tensor fields $$u$$ and $$v$$ satisfy the Lamb equation if $$\partial_tu+\imath_vd_xu=-d_xh$$, where $$\imath_v$$ denotes the interior product, $$h=h(x,t)$$ is a smooth function of its arguments, and the exterior differential $$d_x$$ is taken with respect to the variables $$x$$ only. The author considers the problem of the existence of “relative” and “absolute” integral invariants of the flow $$g^t_v \colon M^n\to M^n$$ of the vector field $$v$$. For example, it is shown that the flow $$g^t_v(x)$$ preserves the value of the integral $$\oint_{g^t_v(\gamma)}u$$, where $$\gamma$$ is a closed curve in $$M^n$$. It can be easily seen that the form $$\Omega=d_xu$$ is preserved by the flow $$g^t_v$$. The form $$u$$ is a relative integral invariant and $$\Omega$$ is an absolute integral invariant.
Chapter III is devoted to the applications of the technique developed in the previous chapters to the case of geodesic flows of left-invariant metrics on Lie groups. The main idea is based on the observation that the Noether integrals, which correspond to the right-invariant Killing symmetries of the left-invariant metric (kinetic energy), give an $$n$$-parameter family of sections of the cotangent bundle $$T^{*}G$$ of the corresponding Lie group $$G$$ that are invariant with respect to the geodesic flow. A hydrodynamic interpretation of the annihilators (Casimir functions) of the Lie-Poisson bracket on the dual space to the corresponding Lie algebra is given.
In Chapter IV the author proposes a vortex variant of the Hamilton-Jacobi method of integration of Hamiltonian systems. Generalizations of the Liouville theorem are given.
Appendix 1 is devoted to the infinite-dimensional Lie group point of view on some equations in hydrodynamics. Appendix 2 considers the connections between hydrodynamics and quantum mechanics.
The book is very interesting and well written. It will be useful for a large number of mathematicians working in different branches of mathematics and physics.

MSC:
 37N10 Dynamical systems in fluid mechanics, oceanography and meteorology 37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory 37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010) 37N05 Dynamical systems in classical and celestial mechanics 53D25 Geodesic flows in symplectic geometry and contact geometry 70H05 Hamilton’s equations 76B47 Vortex flows for incompressible inviscid fluids