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Invariant manifolds of Hamilton’s equations. (English. Russian original) Zbl 1423.37058
J. Appl. Math. Mech. 76, No. 4, 378-387 (2012); translation from Prikl. Mat. Mekh. 76, No. 4, 526-539 (2012).
Summary: The invariance conditions of smooth manifolds of Hamilton’s equations are represented in the form of multidimensional Lamb’s equations from the dynamics of an ideal fluid. In the stationary case these conditions do not depend on the method used to parameterize the invariant manifold. One consequence of Lamb’s equations is an equation of a vortex, which is invariant to replacements of the time-dependent variables. A proof of the periodicity conditions of solutions of autonomous Hamilton’s equations with $$n$$ degrees of freedom and compact energy manifolds that admit of $$2n-3$$ additional first integrals is given as an application of the theory developed.
##### MSC:
 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) 70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics 76A02 Foundations of fluid mechanics
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