Nonlinear Mechanics - Physics at Oregon State University

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12 CHAPTER 1. LAGRANGIAN DYNAMICS Since there are no constraints, the generalized forces are identical with the ordinary forces ℑϕ = − dV dϕ = 0 ℑr = − dV (1.30) dr This equation has an elegant closed form solution in the special case of gravitational attraction. V = − GmM r ≡ −k r (1.31) m¨r = l2 z k − (1.32) mr3 r This apparently nonlinear equation yields to a simple trick, let u = 1/r. d2u dϕ2 + u = m2k l2 z (1.33) If the motion is circular u is constant. Otherwise it oscillates around the value m2k/l2 z with simple harmonic motion. 2 The period of oscillation is identical with the period of rotation so the corresponding orbit is an ellipse. This problem was easy to solve because we were able to discover a nontrivial quantity that was constant, in this case the angular momentum. The constant enabled us to reduce the number of independent variables from two to one. Such a conserved quantity is called an integral of the motion or a constant of the motion. Obviously, the more such quantities one can find, the easier the problem. This raises two practical problems. First, how can we tell, perhaps from looking at the physics of a problem, how many independent conserved quantities there are? Second, how are we to find them? In the central force problem, both of these questions answered themselves. We know that angular momentum is conserved. This fact manifests itself in the Lagrangian in that L depends on ˙ ϕ but not on ϕ. Such a coordinate is said to be cyclic or ignorable. Let q be such a coordinate. Then ( ) d ∂L = 0 (1.34) dt ∂ ˙q The quantity in brackets has a special significance. It is called the canonically conjugate momentum. 3 ∂L ≡ pk (1.35) ˙qk 2 This illustrates a general principle in physics: When correctly viewed, everything is a harmonic oscillator. 3 This notation is universally used, hence the old aphorism that mechanics is a matter of minding your p’s and q’s.
1.5. THE HAMILTONIAN FORMULATION 13 To summarize, if q is cyclic, p is conserved. Suppose we had tried to do the central force problem in cartesian coordinates. Both x and y would appear in the Lagrangian, and neither px nor py would be constant. If we insisted on this, central forces would remain an intractable problem in two dimensions. We need to choose our generalized coordinates so that there are as many cyclic variables as possible. The two questions reemerge: how many are we entitled to and how do we find the corresponding p’s and q’s? A partial answer to the first is given by a well-known result called Noether’s theorem: For every transformation that leaves the Lagrangian invariant there is a constant of the motion. 4 This theorem (which underlies all of modern particle physics) says that there is a fundamental connection between symmetries and invariance principles on one hand and conservation laws on the other. Momentum is conserved because the laws of physics are invariant under translation. Angular momentum is conserved because the laws of physics are invariant under rotation. Despite its fundamental significance, Noether’s theorem is not much help in practical calculations. Granted it gives a procedure for finding the conserved quantity after the corresponding symmetry transformation has been found, but how is one to find the transformation? The physicist must rely on his traditional tools: inspiration, the Ouija Board, and simply pounding one’s head against a wall. The fact remains that there are simple systems, e.g. the Henon-Heiles problem to be discussed later, that have fascinated physicists for decades and for which the existence of these transformations is still controversial. I will have much more to say about the second question. As you will see, there is a more or less “cookbook” procedure for finding the right set of variables and some fundamental results about the sorts of problems for which these procedures are possible. 1.5 The Hamiltonian Formulation I will explain the Hamiltonian assuming that there is only one degree of freedom. It’s easy to generalize once the basic ideas are clear. Lagrangians are functions of q and ˙q. We define a new function of q and p (given by (1.34)). H(p, q) = p ˙q − L(q, ˙q) (1.36) The new function is called the Hamiltonian, and the transformation L → H is called a Lagrange transformation. The equation is much more subtle than 4 See Finch and Hand for a simple proof and further discussion.
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