Solutions To Problems In Goldstein, Classical Mechanics, Second ....pdf ((EXCLUSIVE))
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TY - JOURAU - Gomes, Diogo A.AU - Oberman, AdamTI - Viscosity solutions methods for converse KAM theoryJO - ESAIM: Mathematical Modelling and Numerical AnalysisDA - 2008/9//PB - EDP SciencesVL - 42IS - 6SP - 1047EP - 1064AB - The main objective of this paper is to provenew necessary conditions to the existence ofKAM tori. To do so, we develop a set ofexplicit a-priori estimates for smoothsolutions of Hamilton-Jacobi equations,using a combination of methods fromviscosity solutions,KAM and Aubry-Mather theories.These estimatesare valid in anyspace dimension, and can be checked numericallyto detect gaps between KAM tori and Aubry-Mather sets.We apply these results to detect non-integrable regions in several examples such as a forced pendulum, two coupled penduli, andthe double pendulum. LA - engKW - Aubry-Mather theory; Hamilton-Jacobi integrability; viscosity solutions.; viscosity solutionsUR - -
In 1972, V. Tyulenievich proved that, in each ei- ther Hamiltionian or geometrical setting, the intersection ofthe phase trajectories of the system containing two or more equilibrium points is an invariant set for the flow system. But the conjugacy between the system and the autonomous one can be only local,, or only on the invariant set? This work answers this question for the caseof simple mechanical systems. V. Tyulenievich states that in the case of n- equilibrium points, no complete reduction to autonomous system can be considered, for n>2. In his paper he proves that there exists a reduction to a set of n- equations without any supplementary auxilary conditions. His proof is based on the theory of distribution.. The extension to the case of singular Hamiltonian is also given. Thus we can't consider the reduced system as an autonomous one. For practical reasons this sur- prising result is important. The reduction of trajectory of solves the problem of reduction of the equations of motion to a small movable set, which is the starting point of the application of the method of small movable set in practical problems. In the paper, the long- time asymptotic solutions of the systems of n-equilibrium points are also described. LA - engKW - Distributed control systems; reduction of dynamics; quasirational flows; asymptotic behavior; general bifurcation theoryTI - Inventing a new theory: Conjugacy properties for the Liouville's equationJO - PhD Thesis, LuleåUniversitet, SwedenVL - 47IS - 2SP - 711EP - 11AB - The paper solves the above mentioned question for the case when the numberof equilibrium points is greater than 2. A new condition on the relationships inthe quotient bundles is needed instead of the used one. A new theoremon the global properties of the relation is given. The conditionis formulated in such a way that it allows an easy interpreta- tion of it's meaning, It gives us an necessary condition for theconjugacy between the reduced system of and the autonomous one. The conjugacy is only valid on the invariant set of the reduced system. ed3faa56471