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By A. Jaffe (Chief Editor)

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Variables of the action-angle type on symplectic manifolds stratified by coisotropic tori. Ukr. Math. J. 45, 85–93 (1993) 29. Parasyuk, I. : Reduction and coisotropic invariant tori of Hamiltonian systems with non-Poisson commutative symmetries. I, II, Ukr. Math. J. 46, 572–580, 994–002 (1994) 30. Parasyuk, I. : Coisotropic invariant tori of Hamiltonian systems of the quasiclassical theory of motion of a conduction electron. Ukr. Math. J. 42, 308–312 (1990) 31. : Les m´etodes nouvelles de la m´ecanique c´eleste.

41, 228–238 (1981) 24. : Old and new applications of KAM theory. NATO Conference “Hamiltonian systems of 3 and more degrees of freedom”. Barcelona, August 1995 25. Nekhoroshev, N. : Action-angle variables and their generalizations. Trans. Moscow Math. Soc. 26, 180–198 (1972) 26. : Xn−1 -forming sets of eigenvectors. Proc. Kon. Ned. Akad. Amsterdam 54, 200–212 (1951) 27. Parasyuk, I. : Conservation of multidimensional invariant tori of Hamiltonian systems. Ukr. Math. J. 36, 380–385 (1984) 28. Parasyuk, I.

Up+1 = 0 2 (V, M 2k ). for any p + 1 elements u1 , . . , up+1 ∈ HB Proof. Let the V -invariant closed 2-forms ω1 , . . , ωp+1 represent the elements u1 , 2 . . , up+1 ∈ HB (V, M 2k ). 3). Hence the identity ω1 ∧ · · · ∧ ωp+1 = 0 yields Eq. 4). 11. Concluding Remarks In this paper, we have introduced the concept of integrability of dynamical systems in the broad sense. We have studied the dynamics of an electron on a torus T2 ⊂ R3 in an electromagnetic field and have shown that the corresponding Hamiltonian system on T ∗ (T2 ) is bi-Hamiltonian and integrable in the broad sense.

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