Nonlinear Mechanics - Physics at Oregon State University

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6 CHAPTER 1. LAGRANGIAN DYNAMICS In order to arrive at Poincaré’s moment of discovery, we will have to review the development of classical mechanics through the nineteenth century. This material is found in many standard texts, but I will cover it here in some detail. This is partly to insure uniform notation throughout these lectures and partly to focus on those things that lead directly to chaos in nonlinear systems. We will begin formulating mechanics in terms of generalized coordinates and Lagrange’s equations of motion. We then study Lagrange transformations and use them to derive Hamilton’s equations of motions. These equations are particularly suited to conservative systems in which the Hamiltonian is constant in time, and it is such systems that will be our primary concern. It turns out that Lagrange transformations can be used to transform Hamiltonians in a myriad of ways. One particularly elegant form uses action-angle variables to transform a certain class of problems into a set of uncoupled harmonic oscillators. Systems that can be so transformed are said to be integrable, which is to say that they can be “solved,” at least in principle. What happens, Poincaré asked, to a system that is almost but not quite integrable? The answer entails perturbation theory and leads to the disastrous problem of small divisors. This is the path that led originally to the discovery of chaos, and it is the one we will pursue here. 1.2 Generalized Coordinates and the Lagrangian Vector equations, like F = ma, seem to imply a coordinate system. Beginning students learn to use cartesian coordinates and then learn that this is not always the best choice. If the system has cylindrical symmetry, for example, it is best to use cylindrical coordinates: it makes the problem easier. By “symmetry” we mean that the number of degrees of freedom of the system is less that the dimensionality of the space in which it is imbedded. The familiar example of the block sliding down the incline plane will make this clear. Let’s say that it’s a two dimensional problem with an x-y coordinate system. The block is constrained to move in a straight line, however, so that its position can be completely specified by one variable, i.e. it has one degree of freedom. The clever student chooses the x axis so that it lies along the path of the block. This reduces the problem to one dimension, since y = 0 and the x coordinate is given by one simple equation. In the pendulum example from the previous section, it was most convenient to use a polar coordinate system centered at the pivot. Since r is constant, the motion can be described completely in terms of θ.
1.2. GENERALIZED COORDINATES AND THE LAGRANGIAN 7 These coordinate systems conceal a subtle point: the pendulum moves in a circular arc and the block moves in a straight line because they are acted on by forces of constraint. In most cases we are not interested in these forces. Our choice of coordinates simply makes them disappear from the problem. Most problems don’t have obvious symmetries, however. Consider a bead sliding along a wire following some complicated snaky path in 3-d space. There’s only one degree of freedom, since the particle’s position is determined entirely by its distance measured along the wire from some reference point. The forces are so complicated, however, that it is out of the question to solve the problem by using F = ma in any straightforward way. This is the problem that Lagrangian mechanics is designed to handle. The basic (and quite profound) idea is that even though there may be no coordinate system (in the usual sense) that will reduce the dimensionality of the problem, yet there is usually a system of coordinates that will do this. Such coordinates are called generalized coordinates. To be more specific, suppose that a system consists of N point masses with positions specified by ordinary three-dimensional cartesian vectors, ri, i = 1 · · · N, subject to some constraints. The easiest constraints to deal with are those that can be expressed as a set of l equations of the form fj(r1, r2, . . . , t) = 0, (1.2) where j = 1 · · · l. Such constraints are said to be holonomic. If in addition, the equations of constraint do not involve time explicitly, they are said to be scleronomous, otherwise they are called rheonomous. These constraints can be used to reduce the 3N cartesian components to a set of 3N − l variables q1, q2, . . . , q3N−l. The relationship between the two is given by a set of N equations of the form ri = ri(q1, q2, . . . , q3N−l, t). (1.3) The q’s used in this way are the generalized coordinates. In the example of the bead on a curved wire, the equations would reduce to r = r(q), where q is a distance measured along the wire. This simply specifies the curvature of the wire. It should be noted that the q’s need not all have the same units. Also note that we can use the same notation even if there are no constraints. For example, the position of an unconstrained particle could be written r = r(q1, q2, q3), and the q’s might represent cartesian, spherical, or cylindrical coordinates. In order to simplify the notation, we will often pack the q’s
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