By V. I. Arnol’d, A. B. Givental’ (auth.), V. I. Arnol’d, S. P. Novikov (eds.)

ISBN-10: 3662067935

ISBN-13: 9783662067932

ISBN-10: 3662067951

ISBN-13: 9783662067956

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Extra info for Dynamical Systems IV: Symplectic Geometry and its Applications

Example text

1, chap. 2), a germ of a (co)isotropic submanifold reduces in suitable Darboux coordinates to the linear normal form of sect. 2, chap. 1. The symplectic type of an isotropic submanifold N k c: M 2n in its tubular neighbourhood is determined in a one-one fashion by the equivalence dass ofthe following 2(n - k)-dimensional symplectic vector bundle with base space N k : the fibre of this bundle at a point x is the quotient space of the skew-orthogonal complement to the tangent space TxN by the space TxN itself.

The length of the extremals is a many-valued funetion ofthe point ql with singularities along the rays breaking loose from the obstacle surfaee in an asymptotie direetion. The rays on whieh the extremals issuing from the souree break away from the obstacle surfaee form a Lagrangian variety with singularities in the sympleetie manifold of all the rays of the spaee (eompare seet. 5). The sympleetie analysis of the problem of going around an obstacle leads to the notion of a triad in sympleetie spaee.

At a point x of rank 0 on a Poisson manifold there is weIl defined the linear approximation of the Poisson strueture-a linear Poisson structure on the tangent space of this point (or a Lie algebra structure on the cotangent space): [dxf, dxg] =dx{f, g}. i. B. Givental' The Annihilator Theorem. Let 9 be a Lie algebra, let ~ E 9*, and let 9~ = {x E 91 ad~~ = O} be the annihilator of ~ in 9. The linear approximation of the transversal Poisson strueture to the orbit of the coadjoint action at the point ~ is eanonieally isomorphie to the linear Poisson strueture on the dual spaee of the annihilator 9~.

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Dynamical Systems IV: Symplectic Geometry and its Applications by V. I. Arnol’d, A. B. Givental’ (auth.), V. I. Arnol’d, S. P. Novikov (eds.)


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