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Magnetic flows and Gaussian thermostats on manifolds of negative curvature. (English) Zbl 0997.37011

Let \(M\) be a compact Riemannian manifold with strictly negative sectional curvature. Let \(\pi:TM\to M\) denote the tangent bundle of \(M\), and consider flows on \(TM\) defined by the equations \[ {dq\over dt}= v,\;{Dv\over dt} =-\nabla W+Bv+E -{\langle Ev\rangle \over v^2}v, \] where \(W\) is a smooth function on \(M\), \(\langle , \rangle\) is the inner product, \(B\) is a magnetic field and \(E\) an external field. Here \(D/dt\) denotes the covariant derivative. These flows are further restricted to the level set \(h=\frac 12 v^2+W\), where \(h\) is chosen large enough so that \(v\) does not vanish.
This class of flows includes magnetic flows and Gaussian thermostats, which have been studied by many authors, including Anosov and Sinai, Gouda, Grognet and \(G\). and \(M\). Paternain. In this article the author constructs an explicit quadratic form \(Q\) defined on certain codimension 1 subspaces of the tangent spaces of \(M\), and he shows that the flow is Anosov if the quadratic form \(Q\) is always positive definite. The positivity of \(Q\) has a much simpler form in various special cases, which the author describes.

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
53C80 Applications of global differential geometry to the sciences
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