By L. M. Lerman and Ya. L. Umanskiy

The most subject of this e-book is the isoenergetic constitution of the Liouville foliation generated by means of an integrable approach with levels of freedom and the topological constitution of the corresponding Poisson motion of the crowd ${\mathbb R}^2$. this can be a first step in the direction of knowing the worldwide dynamics of Hamiltonian structures and utilizing perturbation equipment. Emphasis is put on the topology of this foliation instead of on analytic illustration. not like formerly released works during this zone, the following the authors continually use the dynamical homes of the motion to accomplish their effects.

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**Additional resources for Four-dimensional integrable Hamiltonian systems with simple singular points (topological aspects)**

**Sample text**

W e c a ll 7 (h) o r i e n t a b l e o r n o n o r i e n t a b l e h y p e r b o l i c o r b i t , r e s p e c t i v e l y . I t i s c le a r t h a t t h e p r o p e r t y o f b e i n g o r i e n t a b l e o r n o n o r i e n t a b l e is p r e s e r v e d f o r a l l o r b i t s c l o s e e n o u g h t o 7 (h). I t is e a s y t o s e e t h a t f o r a n o r i e n t a b l e h y p e r b o l i c o r b i t , l o c a l l y n e a r 7 ( A ) , t h e c o m m o n le v e l s e t H = h, K = K (l(h )) is a u n i o n o f t w o (h).

Case 13. H (e) = 2^1 ~ 29i 92) - P 192 + |p l , K ( €) = 2 ~ eKfyPi + 2enbpip2 + envp^] ~P\{fbqi -f nvq2) - ^(eK&9i + 2^9i 92 + bql). In Case 14, since A = 0 along the centralizer, we also construct the perturbation of both matrices. The corresponding one-parameter family of perturbations of the Hamilton functions has the form H(e) = ^(9? + «9l) + |(«P? + p |), K (*) = ^9i + ^92 + c(P29i - KP192) + dqiQ2 + y (aPi + 2/tdPiP2 + 6p2), K = ±1. In Case 15 the quantity A also vanishes identically along the centralizer.

Let A be a simple LPA. Let U be its neighborhood in CG 2 and A! G U. There is a matrix IA d in A with simple eigenvalues. Thus, if U is a sufficiently small neighborhood of the point A, then there exists a matrix close to IA d in A' with simple eigenvalues. Indeed, there exists a neighborhood v of the point IA d in sp(4, R) such that all matrices in v have different eigenvalues of the same type as IA d. For any point in CG 2 sufficiently close to A there is a corresponding twodimensional linear subspace in sp(4,R) intersecting v.