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Unique normal forms for vector fields and Hamiltonians. (English) Zbl 0689.70005
The canonical forms we present here yield a countable number of new analytic invariants whose significance for the dynamics needs to be explored on a case by case basis. This task can only be attempted once explicit expressions for these forms become known. On the other hand, it should be no surprise that their calculation is expected to be considerably more involved than that of their classical counterparts. This is because of the additional information that they implicitly carry. In this paper we have only explored the simplest non-trivial examples, namely those of equilibria of planar vector fields with semisimple linear parts. Even in this case the forms encountered are new. For planar rotations the reader is referred to Theorem 6.1 and the dynamical interpretation of some of their coefficients in the remark following that theorem.
Our main results are presented in the setting of graded Lie algebras, a useful medium that allos for a unified treatment of diverse applications to dynamical systems. The necessary prerequisites are standard for vector fields and Hamiltonians; they are collected for the reader’s convenience in Section 1, alongside with generalizations to our context. Sections 2 and 3 contain our main results on specialforms, which are immediately applied in Section 4 to a relatively simple but interesting Lie algebra, here called $${\mathcal N}$$. Section 5 is a partial generalization of the results of Sections 2 and 3 to a larger group that in the context of dynamical systems would include all transformations $$\phi$$ for which $$\phi '(0)$$ is not necessarily the identity. Applications to planar vector fields and to the Hamiltonian 1:1 resonance follow.

##### MSC:
 70H05 Hamilton’s equations 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 17B70 Graded Lie (super)algebras
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##### References:
 [1] Abraham, R; Marsden, J, Foundations of mechanics, (1978), Benjamin-Cummings Reading, MA [2] Baider, A; Churchill, R.C, The Campbell-Hausdorff group and a polar decomposition of graded algebra automorphisms, Pacific J. math., 131, 219-235, (1988) · Zbl 0599.17002 [3] Baider, A; Churchill, R.C, Uniqueness and non-uniqueness of normal forms for vector fields, (), 27-33, in press · Zbl 0671.58010 [4] {\scA. Baider and R. C. Churchill}, Unique normal forms for planar vector fields, Math. Zeits. in press. · Zbl 0691.58012 [5] Bourbaki, N, Commutative algebra, (1973), Hermann Paris, Addison-Wesley, Reading, MA [6] Brjuno, A, Analytical form of differential equations, Trans. Moscow math. soc., 25, 131-288, (1971) [7] Brjuno, A, Analytical form of differential equations, Trans. Moscow math. soc., 26, 199-238, (1972) [8] Churchill, R.C; kummer, M; rod, D.L, On averaging, reduction and symmetry in Hamiltonian systems, J. differential equations, 49, 359-414, (1983) · Zbl 0476.70017 [9] Cushman, R; Sanders, J.A, Nilpotent normal forms and representation theory of sl(2, R), () · Zbl 0712.58053 [10] Kummer, M, How to avoid “secular” terms in classical and quantum mechanics, Nuovo cimenta B, 123-148, (1971) [11] Kummer, M, On resonant non-linearly coupled oscillators with two degrees of freedom, Comm. math. phys., 48, 53-79, (1976) · Zbl 0321.58017 [12] Kummer, M, NSF proposal, (1982) [13] Serre, J.P, Lie algebras and Lie groups, (1965), Benjamin New York [14] Siegel, C.L, On the integrals of canonical systems, Ann. math., 42, 806-822, (1941) · JFM 67.0768.01 [15] Takens, F, Singularities of vector fields, Publ. math. I.H.E.S., 43, 47-100, (1974) · Zbl 0279.58009 [16] van der Meer, J.C, The Hamiltonian Hopf bifurcation, () · Zbl 0853.58053 [17] {\scA. Baider and S. Sanders}, Further reduction of the Takens-Bogdanov normal form, preprint. · Zbl 0761.34027
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