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7 OSCILLATIONS AND CYCLES 54 Observe that if one of the bodies is much bigger than the other (like the Sun and the Earth), say m 1 ≫ m 2 , then the center of mass nearly coincides with the position q 1 of the bigger body, while the reduced mass m is essentially the mass m 2 of the smaller one (hence it looks like the Earth moving around the Sun, as Galileo had suggested). Central forces. Consider the Newton equation m¨r = F (|r|) ˆr describing the motion of a particle (planet) of mass m in a central force field F . Conservation of angular momentum implies that the motion is planar, hence we may take r ∈ R 2 . In polar coordinates r = ρe iθ , the equations reed ¨ρ − ρ ˙θ 2 = F (ρ)/m ρ¨θ + 2 ˙ρ ˙θ = 0 . The second equation says that the “areal velocity” (“velocidade areal”) l = ρ 2 ˙θ is a constant of the motion (Kepler’s second law). Taking Newton’s gravitational force F (ρ) = − GmM ρ (where M is the mass of the Sun and G is 2 the gravitational constant), the first equation may be written as m¨ρ = − ∂ ∂ρ V l (ρ) , where we defined the ”effective potential energy” as V l (ρ) = 1 2 m l2 ρ − G mM 2 ρ is E = 1 2 m ˙ρ2 + 1 2 m l2 ρ 2 − GmM ρ . . The conserved energy Kepler’s effective potential and some energy level sets. Now we set ρ = 1/x and look for a differential equation for x as a function of θ. Computation shows that dx/dθ = − ˙ρ/l, and, using conservation of l, that d 2 x/dθ 2 = −ρ 2 ¨ρ/l 2 . There follows that the first Newton equation reads we get d 2 x dθ 2 + x = − 1 l 2 x 2 m F (1/x) . d 2 x dθ 2 + x = −GM l 2 .
7 OSCILLATIONS AND CYCLES 55 The general solution of this second order linear differential equation is x (θ) = GM l 2 (1 + e cos (θ − θ 0 )) , for some constants e and θ 0 . Back to the original radial variable we get the solution ρ (θ) = l 2 /GM 1 + e cos (θ − θ 0 ) , Hence, orbits are conic sections with eccentricity (“excentricidade” ) e and focus at the origin: an ellipse for 0 ≤ e < 1 (corresponding to negative energy, hence to planets, and this is Kepler’s first law), a parabola for e = 1 (corresponding to zero energy), an hyperbola for e > 1 (corresponding to positive energy). . The 3-body problem and chaos. 18 “Que lon cherche à se représenter la figure formée par ces deux courbes et leurs intersections en nombre infini dont chacune correspond à une solution doublement asymptotique, ces intersections forment une sorte de treillis, de tissu, de réseau à mailles infiniment serrées ; chacune de ces courbes ne doit jamais se recouper elle-même, mais elle doit se replier elle-même dune manière très complexe pour venir couper une infinité de fois toutes les mailles du réseau. On sera frappé de la complexité de cette figure, que je ne cherche même pas à tracer. Rien nest plus propre à nous donner une idée de la complication du problème des trois corps et, en général, de tous les problèmes de dynamique òu il ny a pas d’intégrale uniforme et òu les séries de Bohlin sont divergentes.” 7.4 Cycles in chemistry and biology Henri Poincaré, 1884. 19 Lotka-Volterra predator-prey model. The Lotka-Volterra model is the system of coupled first order differential equations ẋ = ax − bxy ẏ = −cy + dxy defined for positive x and y, where the parameters a, b, c, d are positive constants. These equations, proposed by the physical chemist Alfred Lotka and by the mathematician Vito Volterra between 1925 and 1926, model the population dynamics of two species, x preys and y predator, in the same territory. Preys increase exponentially at rate a and are killed at rate proportional to the probability of beeing captured by a predator, while predators decrease exponentially at rate c and increase at rate proportional to the probability of capturing preys. Discuss the possible dynamics depending on the values of the parameters. Van der Pols. Considere o oscilador de van der Pol 20 ¨q − µ(1 − q 2 ) ˙q + q = 0 que modela a corrente num circuito com um elemento não-linear. 18 A palavra grega χαoς, que pode ser traduzida como “abismo”, contém a mesma base χα- (e probabilmente deriva de) dos verbos χαινειν e χασχειν, que significam ”abrir-se”, “escancarar”, e “abrir a boca”, “bocejar” (cfr. χασµα, “abismo”). Foi utilizada em algumas cosmogonias gregas para indicar “a mistura desordenada de elementos anterior à formação do χoσµoσ, o universo ordenado”. 19 H. Poincaré, Sur le problème de trois corps et les équations de la dynamique, Œuvres, volume VII, Gauthier Villars 1951. 20 B. van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Review 1 (1920), 701-710 and 754-762. B. van der Pol and J. van der Mark, Frequency demultiplication, Nature 120 (1927), 363-364.
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1405R1 - Tópicos de Sistemas Dinâ
Page 3 and 4: CONTEÚDO 3 9 Transversalidade e bi
Page 5 and 6: 1 INTRODUÇÃO 5 Hidrodinâmica. O
Page 7 and 8: 2 ITERATION/RECURSION 7 2 Iteration
Page 9 and 10: 2 ITERATION/RECURSION 9 do primeiro
Page 11 and 12: 2 ITERATION/RECURSION 11 If a is th
Page 13 and 14: 2 ITERATION/RECURSION 13 Nice pictu
Page 15 and 16: 2 ITERATION/RECURSION 15 Experiênc
Page 17 and 18: 3 SISTEMAS DINÂMICOS TOPOLÓGICOS,
Page 19 and 20: 3 SISTEMAS DINÂMICOS TOPOLÓGICOS,
Page 21 and 22: 4 NÚMEROS E DINÂMICA 21 4 Número
Page 23 and 24: 4 NÚMEROS E DINÂMICA 23 4.2 Deslo
Page 25 and 26: 4 NÚMEROS E DINÂMICA 25 Rotaçõe
Page 27 and 28: 4 NÚMEROS E DINÂMICA 27 Observe q
Page 29 and 30: 4 NÚMEROS E DINÂMICA 29 Frações
Page 31 and 32: 5 ÓRBITAS REGULARES E PERTURBAÇÕ
Page 39 and 40: 6 FLOWS 39 6 Flows “... forse sti
Page 41 and 42: 6 FLOWS 41 6.2 Integration of one-d
Page 43 and 44: 6 FLOWS 43 Counter-example. Both th
Page 45 and 46: 6 FLOWS 45 which undergoes a catast
Page 47 and 48: 6 FLOWS 47 Uniqueness of solutons.
Page 49 and 50: 6 FLOWS 49 Dependence on initial da
Page 51 and 52: 7 OSCILLATIONS AND CYCLES 51 7 Osci
Page 53: 7 OSCILLATIONS AND CYCLES 53 Phase
Page 57 and 58: 7 OSCILLATIONS AND CYCLES 57 para a
Page 59 and 60: 8 LINEARIZAÇÃO 59 8 Linearizaçã
Page 61 and 62: 9 TRANSVERSALIDADE E BIFURCAÇÕES
Page 63 and 64: 10 STATISTICAL DESCRIPTION OF ORBIT
Page 75 and 76: 11 RECORRÊNCIAS 75 Exercícios.
Page 77 and 78: 11 RECORRÊNCIAS 77 O conjunto não
Page 79 and 80: 12 TRANSITIVIDADE E ÓRBITAS DENSAS
Page 81 and 82: 12 TRANSITIVIDADE E ÓRBITAS DENSAS
Page 83 and 84: 13 HOMEOMORFISMOS DO CÍRCULO 83 13
Page 85 and 86: 13 HOMEOMORFISMOS DO CÍRCULO 85 de
Page 87 and 88: 14 PERDA DE MEMÓRIA E INDEPENDÊNC
Page 97 and 98: 15 DIMENSIONS, FRACTALS AND ENTROPY
Page 103 and 104: 16 ERGODICITY AND CONVERGENCE OF TI
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16 ERGODICITY AND CONVERGENCE OF TI
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REFERÊNCIAS 107 Referências [AA67
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