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Stability and Chaos

Stability and Chaos. Sir Michael Victor Berry (born 1941). Additional material We will be using two additional sources posted on TCU Online:

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Stability and Chaos

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  1. Stability and Chaos

  2. Sir Michael Victor Berry (born 1941) Additional material • We will be using two additional sources posted on TCU Online: • 1) Berry, M. V., ‘Regular and Irregular Motion’ in Topics in Nonlinear Mechanics, ed. S. Jorna, Am. Inst. Ph. Conf. Proc No.46, pp. 16-120, 1978. (B-1) • 2) Berry, M. V., ‘Semiclassical Mechanics of Regular and Irregular Motion’ in Les Houches Lecture Series Session XXXVI, eds. G. Iooss, R. H. G. Helleman and R. Stora, North Holland, Amsterdam, 1983. (B-2) http://michaelberryphysics.wordpress.com/

  3. (B-2) 2.1 Regular and irregular classical motion • Let us consider dynamics of a classical system with N degrees of freedom described by a Lagrangian (Hamiltonian) – i.e. system without dissipation • It turns out that motion of such a system can be of two types: regular and irregular • Regular motion (very small fraction of all systems in the universe): trajectories with neighboring initial conditions separate linearly • Irregular motion (overwhelming majority of all systems in the universe): trajectories with neighboring initial conditions separate exponentially, resulting in a sensitivity to initial conditions

  4. (B-2) 2.2 (B-1) 2 Integrable systems • The simplest type of a system with a regular motion is an integrable system, for which there exist N smooth independent functions that are constant along the trajectory of the system in the phase space • Such quantities are called constants of motion • For integrable systems, the number of constants of motion is equal to the number of degrees of freedom N

  5. (B-2) 2.2 (B-1) 2 Integrable systems • E.g., for Lagranigans (Hamiltonians) without explicit time dependencies, one of the constants of motion is the total energy (Hamiltonian) • E.g., for systems with central potentials, three constants of motion are the components of the total angular momentum • Along the trajectory: • These equations can be solved for momenta:

  6. (B-2) 2.2 (B-1) 2 Integrable systems • We can consider a canonical transformation to a new set of coordinates Qj and momenta Pj such that the Fj functions to be the new (conserved) momenta • Since the new momenta are constant, the new Hamiltonian will be independent of the generalized coordinates: • Therefore

  7. (B-2) 2.2 (B-1) 2 Integrable systems • The problem is solved if we can express the new generalized coordinates in terms of the old generalized coordinates • We choose a generating function of the form • For such generating function, the Legendre transformations yield • So, the problem is solved since the 2N constants Cj and βj can be found

  8. (B-2) 2.2 (B-1) 2 Integrable systems • The described canonical transformations will work if the conserved quantities Fj satisfy conditions necessary to consider them as independent generalized canonical momenta: • 1) They should be independent (one cannot be derived from the other) • 2) Poisson brackets of all the pairs of the Fj functions vanish • The existence of N functions Fj implies that the system is limited to move on N-dimensional manifold in the 2N-dimensional phase-space

  9. (B-2) 2.2 (B-1) 2 Integrable systems • To describe the motion-restricting N-dimensional manifold we introduce N vector fields in the 2N-dimensional phase space • On each manifold, these vector fields will be smooth and independent (because Fj are smooth and independent)

  10. (B-2) 2.2 (B-1) 2 Integrable systems • The normals to the N-dimensional manifold can be defined as: • On each manifold, the vector fields will be tangential to the manifold (perpendicular to the normals):

  11. (B-2) 2.2 (B-1) 2 11.1 Bounded integrable systems • Let us concentrate of systems with bound motion, i.e. motion with a finite accessible phase space • In this case, the motion-restricting manifold is compact • Topology theorem (without proof): A compact manifold “parallelizable” with N smooth independent fields must be an N-torus:

  12. (B-2) 2.2 (B-1) 2 11.1 Bounded integrable systems • Such tori are called invariant tori, because the orbit starting on such a torus, remains on the torus forever • An N-dimensional torus has N independent irreducible circuitsγj on it • We coordinatize the phase space using Qj and Pj, with {Pj} defining the invariant torus and {Qj} defining the coordinates on the torus • Standard set of such variables: actions and angles on tori

  13. (B-1) 2 11.1 Example: 2D harmonic oscillator • Let us consider a 2D harmonic oscillator: • This Hamiltonian can be separated into two independent Hamiltonians, for each of which • Invariant tori:

  14. (B-1) 2 11.1 Example: 2D harmonic oscillator • If one frequency is a multiple of another, e.g. then the trajectory will close on itself and repeat the same pattern every period • If the frequencies are commensurate (n – rational), then the orbit still will be closed, but will trace out more than one path around p1 and q1 before closing • If the frequencies are incommensurate (n – irrational), then the orbit will never close, gradually covering the entire surface of the torus

  15. (B-1) 2 11.1 Example: 2D harmonic oscillator • For the case of incommensurate frequencies, the orbit is called dense periodic • Dense periodic orbit does not pass through exactly the same point twice, but will eventually pass arbitrarily close to every point • So, the motion is confined to a toroidal surface – a 2D manifold in a 4D phase space

  16. (B-2) 2.3 (B-1) 3 11.2 Nonintegrable systems • Is integrability the rule or the exception? • If all Hamiltonian systems were integrable, the constants of motion would always exist and our inability to determine them for all but the simplest problems would merely reflect our lack of analytical ingenuity • As a matter of fact, there are rigorous analytical and numerical methods showing that most Hamiltonian systems are not integrable, while integrable systems form a very small set (possibly of zero measure) • For nonintegrable systems, the trajectory in phase space fills a region of dimensionality greater than N

  17. (B-2) 2.3 (B-1) 3 11.2 Perturbations • Many Hamiltonian systems can be modeled as a combination of a sum of an integrable system and a small nonintegrable perturbation • Does a small nonintegrable perturbation destroy the tori? • The answer is: it depends • In most cases, the tori persist under small perturbation albeit distorted • Some are destroyed and such tori form a finite set, which grows with the perturbation

  18. (B-1) 3 11.2 12.2 Time-dependent perturbation theory • We start with an unperturbed integrable Hamiltonian • This system has N conserved quantities • We produce canonical transformations employing Hamilton’s principal function • The new Hamiltonian (Kamiltonian) is required to be identically zero, so for the new constant momenta and coordinates we have:

  19. (B-1) 3 11.2 12.2 Time-dependent perturbation theory • The perturbed Hamiltonian is • We use the same functional dependence for S, only now the new canonical variables may not be constant, and the Kamiltonian does not vanish • Now the equations of motion are • These equations generally cannot be solved since the perturbation in nonintegrable

  20. (B-1) 3 11.2 12.2 Time-dependent perturbation theory • Now we take advantage of the smallness of ΔH • In the first order approximation we can write • The notation with zero indicates that after differentiation we substitute αk and βkwith their unperturbed (constant) values • These equations can be written in a symplectic form

  21. (B-1) 3 11.2 12.2 Time-dependent perturbation theory • These equations can be integrated to yield γ1(t) and by inverting the functional dependencies to obtain the time dependencies of qk and pk to the first order of perturbation • The second order perturbation equations are obtained by using γ1(t) dependence in the right hand side: • And for the higher orders:

  22. 12.2 Poisson brackets formalism • Sometimes we need to know the time evolution of some functions of the new canonical set • For the perturbed system, the evolution is described by • We can invert the functional dependence and make K depend on ci • Then:

  23. 12.2 Poisson brackets formalism • If these equations cannot be solved exactly, we apply the perturbation theory approach, i.e. the right-hand-sides are evaluated for the unperturbed motion, etc. • This is a generalization of the perturbation theory equations derived previously

  24. 12.2 Types of perturbed motion • As determined by the perturbation treatment, the parameters of the orbit may vary with time in two ways • There may be a small variation of the orbit around the unperturbed solution, which is not growing with time • There may be a perturbation of the orbit, which is slowly diverging form the unperturbed solution – secular change • The first type of perturbation does not change average parameters of the orbit; the secular perturbation slowly changes orbit parameters

  25. 12.2 Fake example 1: the harmonic oscillator potential as a perturbation • Unperturbed Hamiltonian: • The momentum is conserved • The Hamilton-Jacobi equation:

  26. 12.2 Fake example 1: the harmonic oscillator potential as a perturbation

  27. 12.2 Fake example 1: the harmonic oscillator potential as a perturbation • Let us assume that the small perturbation is • We pretend that we don’t know that the perturbation is integrable • The Kamiltonian is • The perturbed equations of motion:

  28. 12.2 Fake example 1: the harmonic oscillator potential as a perturbation • First order perturbation approximation: • Assuming • Then: • Inverting for x and p:

  29. 12.2 Fake example 1: the harmonic oscillator potential as a perturbation • Second order perturbation approximation: • Solutions: • Inverting for x and p:

  30. 12.2 Fake example 1: the harmonic oscillator potential as a perturbation • If we continue to higher orders, it can be shown that • For • Big surprise!

  31. 12.3 Fake example 2: simple pendulum • Full Hamiltonian: • Expanding: • To obtain a harmonic oscillator approximation, we retain terms with i = 0,1 • Let’s assume that the angles are small, but not small enough to use the harmonic oscillator approximation • Therefore, we have to retain a term with i = 2 as a small perturbation for the harmonic oscillator

  32. 12.3 Fake example 2: simple pendulum • Full (reduced) Hamiltonian: • We assume that we don’t know the solutions of the anharmonic oscillator and accept the anharmonicity as a perturbation • The harmonic oscillator has been solved previously employing the action-angle formalism

  33. 12.3 Fake example 2: simple pendulum • The first order time-dependence: • Averaging over a period Therefore, βis not a constant anymore, but it is changing with time in the following manner: • Averaging over a period • J (which is the measure of the amplitude) does not change with time

  34. (B-1) 3 10.7 12.4 Time-independent perturbation theory • We will consider conservative periodic separable systems with many degrees of freedom and a perturbation parameter ε • For the unperturbed system we introduce a set of action-angle variables {J0i}, {w0i} such that • The original generalized coordinates, which are multiple-periodic in {w0i} (with period unity) can be expanded in the Fourier series of the unperturbed angles

  35. (B-1) 3 10.7 12.4 Time-independent perturbation theory • A compact form of the same Fourier expansion: • The perturbed Hamiltonian can be expanded in the powers of ε: • In the perturbed system, {J0i}, {w0i} remain a valid set of canonical variables, although they are no longer action-angle variables (since the full Hamiltonian depends on {w0i} now), and therefore {J0i} are not constants of motion

  36. (B-1) 3 12.4 Time-independent perturbation theory • If the invariant tori exist in the perturbed system, there must be a new set of action-angle variables {Ji}, {wi} such that • The sets {Ji}, {wi} and {J0i}, {w0i} are related by a canonical transformation generated by a function: • From the Legendre transformation it follows that • Thus the question of the continuing existence of tori reduces to the question of whether the latter equation can be solved (we have to find Y)

  37. (B-1) 3 12.4 Time-independent perturbation theory • We expand both the generating function and the new Hamiltonian in powers of ε: • On the other hand

  38. (B-1) 3 12.4 Time-independent perturbation theory • Using Taylor series expansion:

  39. (B-1) 3 12.4 Time-independent perturbation theory • Using Taylor series expansion:

  40. (B-1) 3 12.4 Time-independent perturbation theory • Therefore:

  41. (B-1) 3 12.4 Time-independent perturbation theory • Thus:

  42. (B-1) 3 12.4 Time-independent perturbation theory • Therefore, for the first order of ε: • If the system remains on the invariant tori, the coordinates and momenta should be periodic functions of {w0i} • Since H1 is a function of q’s and p’s, it is a (given) periodic function of {w0i}

  43. (B-1) 3 12.4 Time-independent perturbation theory • Since Y is a function of q’s and p’s, it is a periodic function of {w0i}; so are all the expansion terms Yk: • The expansion terms Yk are defined up to an arbitrary constant, since Y is a generating function • Let’s choose Y1 in the following form:

  44. (B-1) 3 12.4 Time-independent perturbation theory • Let us consider constant (time-independent) terms: • The remaining terms:

  45. (B-1) 3 12.4 Time-independent perturbation theory • In principle we can continue this algorithm for higher orders in ε • But we have to proceed with caution • What if the unperturbed orbit was closed? • Then we have a case of commensurate frequencies

  46. (B-1) 3 12.4 Time-independent perturbation theory • We can always find such set of ji that • The resonance! (Zero divisors) • The series diverges?! • Actually, there are two concerns about convergence: of the series in powers of ε and the sum

  47. (B-1) 3 12.4 Time-independent perturbation theory • Moreover, even for an open orbit (non-commensurate frequencies), as we go higher and higher of the integer indices in the j-vector we can always find a combination of integers such that • And we have problems with convergence again! • Do all tori get destroyed? • Not so!

  48. (B-1) 3 (B-2) 2.4 11.2 Andrey Nikolaevich Kolmogorov Андрей Николаевич Колмогоров (1903 - 1987) Vladimir Igorevich Arnold Владимир Игоревич Арнольд (1937 - 2010) Jürgen Moser (1928 – 1999) KAM theorem • So, what happens to the perturbed invariant tori? • The answer to this question is given by the celebrated “KAM theorem” (Kolmogorov-Arnold-Moser theorem)

  49. (B-1) 3 (B-2) 2.4 11.2 KAM theorem • KAM theorem (we mentioned this result earlier): “If the bounded motion of an integrable Hamiltonian H0 with N degrees of freedom is disturbed by a small perturbation ΔH, that makes the total Hamiltonian, H = H0 + ΔH, nonintegrable and if two conditions are satisfied: (a) the perturbation ΔH is small (b) the frequencies ωi of H0 are incommensurate, then the motion in 2N-dimensional phase space remains confined to an toroidal manifold of dimension N, except for a negligible set of initial conditions that result in a trajectory on a manifold with a dimension greater than N”

  50. (B-1) 3 KAM theorem • Perturbation theory in the form discussed previously was known and employed for several centuries • However, it turned out that this theory was a verycrude tool for studying the delicate problems arising from the small denominators • The central feature of KAM is the replacement of the series expansions of the conventional perturbation theory by a series of successive approximations to the suspected new tori • This approach has a vastly improved convergence leading to the proof of the theorem

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