Integrability and non-integrability in Hamiltonian mechanics

Abstract

Wagner: Allein die Welt! des Menschen Herz und Geist! Mocht' jeglicher doch was davon erkennen. Faust: Ja, was man so erkennen heisst! Goethe "Faust" CONTENTS 17 §2. Complete integrability 19 §3. Examples of completely integrable systems §4. Perturbation theory §5. Normal forms Chapter Ill. Topological obstructions to complete integrability of natural systems § 1. The topology of the state space of an integrable system § 2. Proof of the theorem on non-integrability §3. Unsolved problems Chapter IV. Non-integrability of nearly integrable Hamiltonian systems § 1. Poincare's method §2. The creation of isolated periodic solutions-an obstruction to integrability § 3. Applications of Poincare's method Chapter V. Bifurcation of asymptotic surfaces §I. Conditions for bifurcation §2. Bifurcation of asymptotic surfaces-an obstruction to integrability 46 §3. Some applications 49 §4. Isolation of the integrable cases 54 Chapter VI. Non-integrability in a neighbourhood of an equilibrium position § 1. Siegel's method §2. Non-integrability of systems depending on a parameter Chapter VII. Branching of solutions and the absence of single-valued integrals § 1. Branching of solutions-an obstruction to integrability § 2. The monodromy groups of Hamiltonian systems with single-valued integrals References

Constants of geodesic motion in higher-dimensional black-hole spacetimes

“…Therefore the constants of motion all Poisson commute (are in involution), so the geodesic motion is completely integrable [16,17].…”

On theC1non-integrability of differential systems via periodic orbits

“…It seems that this result of Poincaré was forgotten in the mathematical community until that modern Russian mathematicians (specially Kozlov) have recently publish on it, see [1,10].…”

Periodic orbits and non-integrability in a cosmological scalar field

“…Now analyzing all the possible combinations, in fact, the possible intersections between the regions R 1 and R 3 ; R 2 and R 4 and the half-lines s ij we obtain the conclusion for the case h > 0. For example, if we take parameters in the region R 1 ∩ R 3 we have 5 periodic orbits and if we take parameters in the half-line s 13 we obtain the existence of only one periodic orbit (see Figure 1). In a similar way we analyze the case h < 0 (see Figure 2).…”

“…The main difficulty for applying the Poincaré non-integrability method to a given Hamiltonian system is to find for such a system periodic orbits having multipliers different from 1. It seems that this result of Poincaré was forgotten by the mathematical community until modern Russian mathematicians (mainly Kozlov) have recently published on it, see [4,13]. Here we will apply the Poincaré criterion to the motion of our cosmological system (5), and we will show that either its motion is integrable and the two constants of motion have dependent gradients along the periodic orbits found in Theorem 1, or it is not Liouville-Arnol'd integrable with any second first integral of class C 1 .…”