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This paper demonstrates sufficient conditions for the existence of Kolmogorov-Arnol’d-Moser (KAM) tori in a singly thermostated, integrable Hamiltonian system with n degrees of freedom with a focus on the generalized, variable-mass thermostats of order 2â€"which include the NosĂ© thermostat, the logistic thermostat of Tapias, Bravetti and Sanders, and the Winkler thermostat. It extends theorem 3.2 of Legoll et al (2009 Nonlinearity 22 1673â€"94) to prove that a ‘typical’ singly thermostated, integrable, real-analytic Hamiltonian possesses a positive-measure set of invariant tori when the thermostat is weakly coupled. It also demonstrates a class of integrable Hamiltonians, which, for a full-measure set of couplings, satisfies the same conclusion. [ABSTRACT FROM AUTHOR]

This note studies 1 and 2 degrees of freedom Hamiltonian systems that are thermostated by a single-variable thermostat. It provides concrete examples of integrable Hamiltonians and single thermostats, including the Nos\'e--Hoover thermostat, where the existence of a horseshoe in the flow of the thermostated system is proven. [ABSTRACT FROM AUTHOR]

This paper studies the dynamics of a family of Hamiltonian systems that originate from Friedman–Lemaître–Robertson–Walker space-times with a coupled field and non-zero curvature. In four distinct cases, previously considered by Maciejewski, Przybylska, Stachowiak and Szydowski, it is shown that there are homoclinic connections to invariant submanifolds and the connections split. These results imply the non-existence of a real-analytic integral independent of the Hamiltonian. [ABSTRACT FROM AUTHOR]

This paper demonstrates sufficient conditions for the existence of a positive measure set of invariant KAM (Kolmogrov-Arnol'd-Moser) tori in a singly thermostated, one degree-of-freedom Hamiltonian vector field. This result is applied to four important single thermostats in the literature, and it is shown that in each case, if the Hamiltonian is real-analytic and well-behaved, then the thermostated system always has a positive measure set of invariant KAM tori for sufficiently weak coupling and high temperature. This extends the results of Legoll, Luskin, and Moeckel [Arch. Ration. Mech. Anal. 184, 449 (2007)] and Legoll et al. [Nonlinearity 44, 1673 (2009)]. [ABSTRACT FROM AUTHOR]

Let $${H(q,p) = \frac{1}{2}{\parallel p \parallel}^2 + V(q)}$$ be an n-degree of freedom C mechanical Hamiltonian on $${T^{*}{\bf T}^n}$$ where $${r > 2n+2}$$ . When the metric $${{\parallel \cdot \parallel}}$$ is flat, the Nosé-thermostated system associated to H is shown to have a positive-measure set of invariant tori near the infinite temperature limit. This is shown to be true for all variable mass thermostats similar to Nosé's, too. These results complement results of Bulter (Nonlinearity 11(29):3454-3463, 2016), Legoll et al. (Arch Ration Mech Anal 184(3):449-463, 2007, Nonlinearity 22(7):1673-1694, 2009). [ABSTRACT FROM AUTHOR]

This paper studies the existence of invariant smooth Lagrangian graphs for Tonelli Hamiltonian systems with symmetries. In particular, we consider Tonelli Hamiltonians with n independent but not necessarily involutive constants of motion and obtain two theorems reminiscent of the Liouville-Arnol′d theorem. Moreover, we also obtain results on the structure of the configuration spaces of such systems that are reminiscent of results on the configuration space of completely integrable Tonelli Hamiltonians. [ABSTRACT FROM AUTHOR]

This paper proves two main results. First, it is shown that if Σ is a smooth manifold homeomorphic to the standard n-torus Tn = Rn/Zn and H is a real-analytically completely integrable convex hamiltonian on T*Σ, then Σ is diffeomorphic to Tn. Second, it is proven that for some topological 7-manifolds, the cotangent bundle of each smooth structure admits a real-analytically completely integrable riemannian metric hamiltonian. [ABSTRACT FROM AUTHOR]

We consider magnetic flows on compact quotients of the 3-dimensional solvable geometry Sol determined by the usual left-invariant metric and the distinguished monopole. We show that these flows have positive Liouville entropy and therefore are never completely integrable. This should be compared with the known fact that the underlying geodesic flow is completely integrable in spite of having positive topological entropy. We also show that for a large class of twisted cotangent bundles of solvable manifolds every compact set is displaceable. [ABSTRACT FROM AUTHOR]

This paper studies completely integrable hamiltonian systems onT*S where S is abundle overwith an R-split, free abelian monodromy group. For each periodic Toda lattice there is an integrable hamiltonian system onT*S with positive topological entropy. Bolsinov and Taimanov’s example of an integrable geodesic flow with positive topological entropy fits into this general construction with theA(1)1 Toda lattice. Topological entropy is used to show that the flows associated to non-dual Toda lattices are typically topologically non-conjugate via an energy-preserving homeomorphism. The remaining cases are approached via the homology spectrum. An energy-preserving conjugacy implies the congruence of two rational quadratic forms over the unit group of a number fieldF. WhenF/Q is normal a classification of flows is obtained. In degree 3, this results from a well-known result of Gelfond; in higher degrees, the result is conditional on the conjecture that a rationally independent set of logarithms of algebraic numbers is algebraically independent over Q. [ABSTRACT FROM AUTHOR]