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

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

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The Foliation Theorem ([42], [74]). The equivalence dass of an arbitrary point of a Poisson manifold is a symplectic submanifold of a dimension equal to the rank of the Poisson strueture at that point. Thus, a Poisson manifold breaks up into sympleetie leaves, which in aggregate determine the Poisson structure: the Poisson bracket of two functions can be computed over their restrictions to the symplectic leaves. A transversal to a symplectic leaf at any point intersects the neighbouring symplectic leaves transversally along symplectic manifolds and inherits a Poisson structure in a neighbourhood of the original point.

X k = 0 be the equations of X and let Yl' . . , Y2n-k be the remaining coordinates on [R2n. Since the form W l -Wo is equal to zero on X, then IY. = L (XilY. i + /;dx i ) + df, where the lY. i are 1-forms and the /; and f are functions,f depending only on y. Consequently, we may replace IY. by the form LXi(lY. i - d/;) = IY. - dU + I/;x;), equal to zero at the points of X. 4. The Classification of Submanifold Germs. The relative Darboux theorem allows us to transform information on the degeneracies of closed 2forms into results on the classification of germs of submanifolds 01' symplectic space.

C) The Poisson theorem: The Poisson bracket of two first integrals of a H amiltonian fiow is again a first integral. Example. If in a mechanical system two components M l' M 2 of the angular momen turn vector are conserved, then also the third one M 3 = {M l' M 2} will be conserved. 2. Poisson Manifolds. A Poisson structure on a manifold is abilinear form { ,} on the space of smooth functions on it satisfying the requirement of anticommutativity, the Jacobi identity and the Leibniz rule (see corollary 2 ofthe preceding item).

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