Effective Stability of the equilateral Lagrangepoints

Contrary to the KAM theorem (Kolmogorov 1954, Arnold 1963, Moser 1962), which asserts stability for all times of those orbits of a nonlinear dynamical system, with initial conditions belonging to a Cantor set of tori of non-zero measure, exponential stability is of much greater interest from the physical point of view, because it can be applied to all orbits in open domains of the phase space, whether they lie on an invariant torus or not. The corresponding theorem proven by Nekhoroshev (1977) defines stability regions for a finite time T in both, regular and chaotic domains of the phase space. If the life-time of the physical system is shorter than the stability time derived from the Nekhoroshev estimates of the region, one can definitely say that orbits belonging to this region are stable from the practical point of view. An already known example of the Lagrangian equilateral configuration in our solar system, where Nekhoroshev estimates are applicable, is the population of Trojan asteroids around the equilibrium points L4 and L5 of the Sun-Jupiter system. The Trojans are in the 1:1 mean motion resonance with Jupiter and their motions and semimajor axes oscillate around those of Jupiter as mean values. While moving around the Sun, the Trojans perform librating motions around Jupiter`s L4 or L5 point, located at 60 before and after Jupiter on Jupiter`s orbit. All investigations of the Nekhoroshev stability of the Trojan type motion up to now were based on the circular restricted three body problem. This circular model is a great simplification of the real physical system that neglects all phenomena due to the ellipticity of the orbit of Jupiter and its secular changes due to the presence of the other planets. Therefore it is a mathematical challenge and represents a physically very interesting problem to generalize the Nekhoroshev estimates of the size of the stable region. Based on the method of Hadjimetriou we will develop a set of corresponding 4 dimensional mappings of the Trojan-type motion for different eccentricities of the perturbing body to make our results applicable also to Trojan planets in exosolar systems. Using the theory of Birkhoff normal forms, we will find an explicit form of the remainder of the mappings, which will directly lead us to suitable Nekhoroshev estimates of the Trojan-type motion. Thus our work is a logical next step on getting closer to more realistic applications of Nekhoroshev theory in celestial mechanics.

Contrary to the KAM theorem (Kolmogorov 1954, Arnold 1963, Moser 1962), which asserts stability for all times of those orbits of a nonlinear dynamical system, with initial conditions belonging to a Cantor set of tori of non-zero measure, exponential stability is of much greater interest from the physical point of view, because it can be applied to all orbits in open domains of the phase space, whether they lie on an invariant torus or not. The corresponding theorem proven by Nekhoroshev (1977) defines stability regions for a finite time T in both, regular and chaotic domains of the phase space. If the life-time of the physical system is shorter than the stability time derived from the Nekhoroshev estimates of the region, one can definitely say that orbits belonging to this region are stable from the practical point of view. An already known example of the Lagrangian equilateral configuration in our solar system, where Nekhoroshev estimates are applicable, is the population of Trojan asteroids around the equilibrium points L4 and L5 of the Sun-Jupiter system. The Trojans are in the 1:1 mean motion resonance with Jupiter and their motions and semimajor axes oscillate around those of Jupiter as mean values. While moving around the Sun, the Trojans perform librating motions around Jupiter`s L4 or L5 point, located at 60 before and after Jupiter on Jupiter`s orbit. All investigations of the Nekhoroshev stability of the Trojan type motion up to now were based on the circular restricted three body problem. This circular model is a great simplification of the real physical system that neglects all phenomena due to the ellipticity of the orbit of Jupiter and its secular changes due to the presence of the other planets. Therefore it is a mathematical challenge and represents a physically very interesting problem to generalize the Nekhoroshev estimates of the size of the stable region. Based on the method of Hadjimetriou we will develop a set of corresponding 4 dimensional mappings of the Trojan-type motion for different eccentricities of the perturbing body to make our results applicable also to Trojan planets in exosolar systems. Using the theory of Birkhoff normal forms, we will find an explicit form of the remainder of the mappings, which will directly lead us to suitable Nekhoroshev estimates of the Trojan-type motion. Thus our work is a logical next step on getting closer to more realistic applications of Nekhoroshev theory in celestial mechanics.