Global surfaces of section in non-regular convex energy levels of Hamiltonian systems

In this paper we prove the existence of global sections of disk-type in non-regular and strictly convex energy levels of integrable and near-integrable Hamiltonian systems with two degrees of freedom. This extends a result of (Hofer et al. in Ann. Math.(2) 148(1):197–289, 1998) where the same statement is true provided the energy level is regular.     

Existence of even homoclinic orbits for second-order Hamiltonian systems

Some existence theorems for even homoclinic orbits are obtained for a class of second-order nonautonomous Hamiltonian systems with symmetric potentials under a class of new superquadratic conditions. A homoclinic orbit is obtained as a limit of solutions of a certain sequence of nil-boundary-value problems which are obtained by the minimax methods.     

On Anosov energy levels of convex Hamiltonian systems

Supported by the Max-Planck-Institut für Mathematik and by a travel grant from CDE. On leave from: IMERL-Facultad de Ingenieria, Julio Herrera y Reissig 565, Montevideo-Uruguay     

The compact category and multiple periodic solutions of Hamiltonian systems on symmetric starshaped energy surfaces

Dieter Puppe zum 60. Geburtstag gewidmet     

Existence of periodic solutions of Hamiltonian systems on almost every energy surface

We estab lish the existence of periodic solutions of Hamiltonian systems on almost every smooth, compact energy surface in the sense of Lebesgue measure.     

Existence theorems for regular,closed, convex surfaces

With the help of C. Miranda's method, developed in RZh. Mat. 1972, IA 1121 and 2A 917, existence problems are studied for closed convex surfaces whose principal radii of curvatureR ₁(n) andR ₂(n) satisfy an equation of the form R₁R₂ + (R₁ + R₂, R₁, R₂, n) + cn = (n), where c is a constant vector connected to the desired surface and the closure condition holds for(n). Here, in contrast to C. Miranda's papers, it is not assumed that ₁0. Instead, it is required that the first partial derivatives of with respect toR ₁ andR ₂ be nonnegative. A special case of the proved general theorem is the theorem about the existence of an equation in which is equal to the reciprocal of the mean curvature of the surface. The question of carrying over certain of Miranda's results to the case where increases as (R₁R₂)^µ, where µ>1, is also considered.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 69–80, 1991.     

Generalized convex functions on normal decomposition systems

Aequationes mathematicae - In this paper, the class of weakly $$ \psi $$ -uniformly convex functions is introduced and studied. Such functions are defined on a convex subcone D of a linear space V...     

Singularities of Blaschke normal maps of convex surfaces

We prove that the difference between the numbers of positive swallowtails and negative swallowtails of the Blaschke normal map for a given convex surface in affine space is equal to the Euler number of the subset where the affine shape operator has negative determinant.     

Existence of multiple periodic orbits on star-shaped hamiltonian surfaces

Consider the Hamiltonian system (HS) i = 1, …, N. Here, H ? C²(?^2N, ?). In this paper, we investigate the existence of periodic orbits of (HS) on a given energy surface Σ = {z ? ?^2N; H(z) = c} (c > o is a constant). The surface Σ is required to verify certain geometric assumptions: Σ bounds a star-shaped compact region ? and α? ? ? ? β? for some ellipsoid ? ? ?^2N, o < α < β. We exhibit a constant δ > O (depending in an explicit fashion on the lengths of the main axes of ? and one other geometrical parameter of Σ) such that if furthermore β²/α² < 1 + δ, then (HS) has at least N distinct geometric orbits on Σ. This result is shown to extend and unify several earlier works on this subject (among them works by Weinstein, Rabinowitz, Ekeland-Lasry and Ekeland). In proving this result we construct index theories for an S¹ -action, from which we derive abstract critical point theorems for S¹ -invariant functionals. We also derive an estimate for the minimal period of solutions to differential equatious.     

Multiple periodic orbits of asymptotically linear autonomous convex Hamiltonian systems

We investigate multiple periodic solutions of convex asymptotically linear autonomous Hamiltonian systems. Our theorem generalizes a result by Mawhin and Willem.     

Additional integrals of the motion of classical Hamiltonian wave systems

Institute of Water Problems, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 76, No. 1, pp. 88–99, July, 1988.     

Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems

By using the Symmetric Mountain Pass Theorem, we establish some existence criteria to guarantee the second-order self-adjoint discrete Hamiltonian system Δ[p(n)Δu(n−1)]−L(n)u(n)+∇W(n,u(n))=0 has infinitely many homoclinic orbits, where n∈Z, u∈RN, p,L:Z→RN×N and W:Z×RN→R are no periodic in n. Our conditions on the potential W(n,x) are rather relaxed.     

Existence of homoclinic solution for the second order Hamiltonian systems

An existence theorem of homoclinic solution is obtained for a class of the nonautonomous second order Hamiltonian systems , ∀t∈R, by the minimax methods in the critical point theory, specially, the generalized mountain pass theorem, where L(t) is unnecessary uniformly positively definite for all t∈R, and W(t,x) satisfies the superquadratic condition W(t,x)/|x|²→+∞ as |x|→∞ uniformly in t, and need not satisfy the global Ambrosetti-Rabinowitz condition.