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What Are Completely Integrable Hamilton Systems

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What Are Completely Integrable Hamilton Systems THE TEACHING OF MATHEMATICS 2011, Vol. XIII, 1, pp. 1–14 WHAT ARE COMPLETELY INTEGRABLE HAMILTON SYSTEMS BoˇzidarJovanovi´c Abstract. This paper is aimed to undergraduate students of mathematics and mechanics to draw their attention to a modern and exciting field of mathematics with applications to mechanics and astronomy. We cared to keep our exposition not to go beyond the supposed knowledge of these students. ZDM Subject Classification: M55; AMS Subject Classification: 97M50. Key words and phrases: Hamilton system; completely integrable system. 1. Introduction Solving concrete mechanical and astronomical problems was one of the main mathematical tasks until the beginning of XX century (among others, see the work of Euler, Lagrange, Hamilton, Abel, Jacobi, Kovalevskaya, Chaplygin, Poincare). The majority of problems are unsolvable. Therefore, finding of solvable systems and their analysis is of a great importance. In the second half of XX century, there was a breakthrough in the research that gave a basis of a modern theory of integrable systems. Many great mathemati- cians such as P. Laks, S. Novikov, V. Arnold, J. Mozer, B. Dubrovin, V. Kozlov contributed to the development of the theory, which connects the beauty of classi- cal mechanics and differential equations with algebraic, symplectic and differential geometry, theory of Lie groups and algebras (e.g., see [1–4] and references therein). The aim of this article is to present the basic concepts of the theory of integrable systems to readers with a minimal prior knowledge in the graduate mathematics, so we shall not use a notion of a manifold, symplectic structure, etc. The most of mechanical and physical systems are modelled by Hamiltonian equations. We shall introduce Hamiltonian systems with one of the easiest prob- lems, the system of n independent harmonic oscillators (Section 2). The corre- sponding equations, as linear, are easily solvable. The solutions are expressed as a trigonometric functions of time and the dynamics is linearized on invariant tori. It turns out that much more complex Hamiltonian problems, if they have a sufficient number of integrals of motion (conservation laws), have qualitatively a similar behavior. It is a content of the Liouville-Arnold theorem which we state without a proof in Section 4. In Section 3, we define the canonical Poisson brack- ets. This is a very important geometric structure that makes possible to study Hamiltonian systems using analytical, algebraical and geometrical methods. 2 B. Jovanovi´c Finally in Section 5, the system of two identical harmonic oscillators is consid- ered. It is shown that this system provides a natural description of an important geometrical object, the Hopf fibration. 2. Hamiltonian equations and harmonic oscillators 2.1. We have learned already in a high school about the harmonic oscillator, describing the motion of a material point under the influence of a force ¡aq, a > 0, where q is a shift from the equilibrium position q = 0 (for example, a motion of a material point related to the elastic spring with the elastic coefficient a). The system is described by the equation: (1) mq¨ = ¡aq; where m is the mass of a material point. By introducing of a new variable p = mq˙ (the moment) and the function 1 2 a 2 h(q; p) = 2m p + 2 q (the sum of kinetic and potential energy of the system), the second order equation (1) takes the form of the system of two first order equations in the space R2(q; p): p @h @h (2)q ˙ = = ; p˙ = ¡aq = ¡ : m @p @q Fig. 1. The system of n independent harmonic oscillators Now, let us consider the system of n independent harmonic oscillators (e.g., the system of n elastic springs). As above, we can write equations of motion in the 2n 2n space R (q; p) = R (q1; : : : ; qn; p1; : : : ; pn): pi @h @h (3)q ˙i = = ; p˙i = ¡aiqi = ¡ ; i = 1; : : : ; n mi @pi @qi Pn 2 2 (mi; ai > 0), where h(q; p) = i=1(pi =mi +aiqi )=2 represents the total mechanical energy of the system. 2.2. The system of equations @h @h (4)q ˙i = ; p˙i = ¡ ; i = 1; : : : ; n @pi @qi 2n defined in a region U of the space R (q1; : : : ; qn; p1; : : : ; pn) is called a Hamiltonian system. The function h is called Hamiltonian, while the region U is called the phase space of the system. By introducing the variable x = (q1; : : : ; qn; p1; : : : ; pn) and a Hamiltonian vector field µ ¶ @h @h @h @h (5) Xh = ;:::; ; ¡ ;:::; ¡ ; @p1 @pn @q1 @qn system (4) reads simplyx ˙ = Xh(x). What are integrable Hamilton systems 3 2.3. Let us recall some basic notions from the theory of ordinary differential equations. Consider the equations n (6)x ˙ = X(x); x = (x1; : : : ; xd) 2 U ½ R ; where X = (X1(x);:::;Xd(x)) is a smooth vector field on U. By the theorem on existence and uniqueness of solutions, through each point x0 2 U and a moment of 0 time t0, there is a unique solution x(t), such that x(t0) = x . Under integrability of the system (6), in broadest context, we mean the finding 0 of the solutions x(t) for a general initial condition x(t0) = x . The most important objects in solving the equation are (first) integrals of the motion. A function F is a first integral of equation (6) if it is constant along solutions: f(x(t)) = const. It is clear that f is an integral if and only if the derivative of f in the direction vector field X identically equals to zero: Xd @f X(f) = X ´ 0: i @x i=1 i Geometrically, this means that if at some point of time t0 a trajectory x(t) belongs to Mc = ff(x) = cg, then it lies within Mc for every t. If the differential df is different from zero on Mc, then Mc is a smooth hyper-surface. The vectors X(x) are tangent to Mc for every x 2 Mc. Fig. 2. Invariant surface Now, if equation (6) have l functionally independent integrals, the problem re- duces (for general values of the parameters c = (c1; : : : ; cl)) to solving the restricted system on (d ¡ l)-dimensional smooth invariant surface Mc : f1(x) = c1; : : : ; fr(x) = cl: It is clear that the existence of d ¡ 1 integrals implies solvability of (6). 4 B. Jovanovi´c 2.4. As we shall see soon, in the framework of Hamiltonian mechanics, for solving equation (4) there should be “only” n integrals. Even more, the phase space of a completely integrable Hamiltonian system has a very fine structure: it is, almost everywhere, decomposed on invariant n-dimensional tori in which the dynamics is linearized. We mentioned that we shall skip using of manifolds. However, there is one exception. The main geometrical object in the theory of integrable systems is a torus, and we hope readers will become familiar with its use. A torus (n-dimensional) Tn is a direct product of n circles: Tn = S1 £ ¢ ¢ ¢ £ S1 : | {z } n n Let ('1;:::;'n) be linear coordinates of the vector space R . Then we have a n n natural mapping ¼ : R ! T , ¼(') = ' mod 2¼, where 'i (mod 2¼) is the angular variable on the i-th circle. The mapping ¼ is 2¼-periodic in each variable. n n Thus, we can represent torus as an n-dimensional cube [0; 2¼] ½ R ('1;:::;'n) with identified opposite sides of the cube: ('1;:::;'i¡1; 0;'i+1;:::;'n) ´ ('1;:::;'i¡1; 1;'i+1;:::;'n); i = 1; : : : ; n: A linear (quasi-periodic, conditionally periodic) motion on Tn is a projection of a line with the mapping ¼: 0 (7) 'i(t) = 'i + !it; i = 1; : : : ; n: The numbers !1;:::;!n are called frequencies. The trajectory (7) is called a wind- ing of the torus, as well. Fig. 3. Linear trajectories on a two-dimensional torus for !2=!1 2= Q, !2=!1 ¼ 3=2 (non-periodic) and !2=!1 = 3=2 (periodic), respectively. 2.5. One can solve (2) easily. First note that the energy h is an integral: d 1 a 1 1 h(q; p) = 2pp˙ + 2qq˙ = apq ¡ apq = 0: dt 2m 2 m m What are integrable Hamilton systems 5 2 The equation h(q; p) = E, for E > 0 defines the ellipse ME in R (q; p). Let us introduce an angular variable ' (mod 2¼) on ME (see Fig. 2): p p (8) q = 2E=a cos('); p = ¡ 2Em sin('): p From the equations of motion (2) we get' ˙ = a=m. Hence, by integration, we obtain the well known expressions p ³p ´ p ³p ´ q(t) = 2E=a cos a=mt + '0 ; p(t) = ¡ 2Em sin a=mt + '0 : The constants E ¸ 0 (energy) and '0 2 [0; 2¼) are determined from the initial con- dition (q(t0); p(t0)). The initial condition (q(t0); p(t0)) = (0; 0) gives the equilibrium position, namely the solution is (q(t); p(t)) ´ (0; 0). Fig. 4. The phase space of a harmonic oscillator 2.6. System (3) has n independent integrals: 1 2 ai 2 (9) fi(q; p) = fi(qi; pi) = pi + qi ; i = 1; : : : ; n 2mi 2 (the energy of i-th harmonic oscillator) and the phase space is foliated on invariant surfaces 2n (10) Mc = f(q1; : : : ; qn; p1; : : : ; pn) 2 R j f1 = c1; : : : ; fn = cng: In general, if all constants ci are greater then zero, Mc is a product of ellipses: Mc = Ec1 £¢ ¢ ¢£Ecn , Eci = f(qi; pi) j fi = cig and represents an n-dimensional torus n T . If one of the constants ci is zero, the invariant manifold Mc is less dimensional torus.
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