Nonlinear Science

libations, or periodic oscillations. The open, “wavy” lines away from the hori- By comparing Eqs. 1 and 3 and recalling unbounded motions in the sense that 6’ in- creases or decreases forever as the pendu- lum rotates around its pivot point in either That the constant C is proportional to the value of the Hamiltonian, of course, is What about other systems? A dy- just an expression of the familiar conser- namical system that can be described by The Simple vation of energy and shows that the value 2N generalized position and momentum of the conserved energy determines the coordinates is said to have N degrees but NONLINEAR nature of the pendulum’s motion. of freedom. Hamiltonian systems that, Restricting our considerations to libra- PENDULUM like the pendulum, have only one de- tions—that is, motions in which the pen- gree of freedom can always be integrated dulum oscillates back and forth without Elementary physics texts typically treat completely with the techniques used for swinging over the top of its pivot point— the simple plane pendulum by solving the Eqs. 2-6. More generally, however, sys- equation of motion only in the linear ap- tems with N degrees of freedom are not proximation and then presenting the gen- completely integrable; Hamiltonian sys- which yields eral solution as a superposition of sines tems with N degrees of freedom that and cosines (as in Eq. 3 of the main text). are completely integrable form a very re- However, the full nonlinear equation can This, in turn, means that stricted but extremely important subset of also be solved analytically in closed form, all N-degree-of-freedom systems. and a brief discussion of this solution al- As suggested by the one-degree-of- lows us to illustrate explicitly several as- freedom case, complete integrability of pects of nonlinear systems. The full period of the motion T is then a system with N degrees of freedom re- It is most instructive to start our anal- the definite integral quires that the system have N constants ysis using the Hamiltonian for the sim- of motion—that is, N integrals analo- ple pendulum, which, in terms of the . (6) gous to Eq. 4-and that these constants o be consistent with each other. Techni- the corresponding (generalized) momen- This last integral can be converted, via cally, this last condition is equivalent to trigonometric identities and redefinitions saying that when the constants, or inte- of variables, to an elliptic integral of the grals of motion, are expressed in terms first kind. Although not as familiar as of the dynamical variables (as C is in the sines and cosines that arise in the Eq. 4), the expressions must be “in invo- Using the Hamiltonian equations linear approximation, the elliptic integral lution,” meaning that the Poisson brack- is tabulated and can be readily evalu- ets must vanish identically for all possible ated. Thus, the full equation of motion pairs of integrals of motion. Remarkably, for the nonlinear pendulum can be solved one can find nontrivial examples of com- in closed form for arbitrary initial condi- pletely integrable systems, not only for tions. N-degree-of-freedom systems but also for An elegant method for depicting the the “infinite’ ’-degree-of-freedom systems solutions for the one-degree-of-freedom described by partial differential equations. and its derivative: system is the “phase plane.” If we ex- The sine-Gordon equation, discussed ex- amine such a plot (see Fig. 2 in the main tensively in the main text, is a famous example. In spite of any nonlinearities, systems that are completely integrable possess re- represent the pen- markable regularity, exhibiting smooth can be converted to a perfect differential stable fixed points with dulum at rest and the bob pointing down. motion in all regions of phase space. This fact is in stark contrast to nonintegrable systems. With as few as one-and-a-half degrees of freedom (such as the damped, ble fixed points with the pendulum at rest driven system with three generalized co- but the bob inverted; the slightest pertur- ordinates represented by Eq. 4 in the main bation causes the pendulum to move away text), a nonintegrable system can exhibit Hence, we can integrate Eq. 3 immedi- from these points. The closed curves near deterministic chaos and motion as random ately to obtain as a coin toss. ■

Los Alamos Science Special Issue 1987 221