Nonlinear Mechanics - Physics at Oregon State University

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22 CHAPTER 2. CANONICAL TRANSFORMATIONS 2.3 Hamilton’s Principle Function The F1 style generating functions were used to transform to a new set of variables (q, p) → (Q, P ) such that all the Q’s were cyclic. As a consequence, the P ’s were constants of the motion, and the Q’s were linear functions of time. The generating function itself was hard to find, however. The F2 generating function goes one step further; it can transform to a set of variables in which both the Q’s and P ’s are constant and simple functions of the initial values of the phase space variables. In essence, our transformation is (q(t), p(t)) ↔ (q0, p0) This is a time dependent transformation, of course. The fact that we can find such transformations shows that the time evolution of a system is itself a canonical transformation. We look for an F2 so that K in (2.18) is identically zero! Then from (2.6), ˙ Qk = 0 and ˙ Pk = 0. The appropriate generating function will be a solution to H(q, p, t) + ∂F2 ∂t We eliminate pk using (2.16) ( H q1, . . . , qn; ∂F2 , . . . , ∂q1 ∂F2 ) ; t + ∂qn ∂F2 ∂t (2.19) = 0. (2.20) The solution to this equation is usually called S, Hamilton’s principle function. The equation itself is the Hamilton-Jacobi equation. 2 There are two serious issues here: does it have a solution, and if it does, can we find it? We will take a less serious approach: if we can find a solution, then it most surely exists. Furthermore, if we can find it, it will have the form S = ∑ Wk(qk) − αt (2.21) k Partial differential equations that have solutions of the form (2.21) are said to be separable. 3 Most of the familiar textbook problems in classical mechanics and atomic physics can be separated in this form. The question of separability does depend on the system of generalized coordinates used. For example, the Kepler problem is separable in spherical coordinates but not in cartesian coordinates. It would be nice to know whether a particular 2 See Goldstein, Classical Mechanics, Chapter 10 3 Or to be meticulous, completely separable
2.3. HAMILTON’S PRINCIPLE FUNCTION 23 Hamiltonian could be separated with some system of coordinates, but no completely general criterion is known. 4 As a rule of thumb, Hamiltonians with explicit time dependence are not separable. If our Hamiltonian is separable, then when (2.21) is substituted into (2.20), the result will look like f1 ( q1, dW1 dq1 ) + f2 ( q2, dW2 ) + · · · = α (2.22) dq2 Each function fk is a function only of qk and dWk/dqk. Since all the q’s are independent, each function must be separately constant. This gives us a system of n independent, first-order, ordinary differential equations for the Wk’s. ( fk qk, dWk ) = αk. (2.23) dqk The W ’s so obtained are then substituted into (2.21). The resulting function for S is F2 ≡ S = S(q1, . . . , qn; α1, . . . , αn; α, t) The final constant α is redundant for two reasons: first, ∑ αk = α, and second, the transformations equations (2.16) and (2.17) involve derivatives with respect to qk and Pk. When S is so differentiated, the −αt piece will disappear. In order to make this apparent, we will write S as follows: F2 ≡ S = S(q1, . . . , qn; α1, . . . , αn; t) (2.24) Since the F2 generating functions have the form F2(q, P ), we are entitled to think of the α’s as “momenta,” i.e. αk in (??) corresponds to Pk in (2.17). In a way this makes sense. Our goal was to transform the time-dependent q’s and p’s into a new set of constant Q’s and P ’s, and the α’s are most certainly constant. On the other hand, they are not the initial momenta p0 that evolve into p(t). The relationship between α and p0 will be determined later. If we have dome our job correctly, the Q’s given by (2.17) are also constant. They are traditionally called β, so Qk = βk = ∂S(q, α, t) ∂αk Again, β’s are constant, but they are not equal to q0. We can turn this into a cookbook algorithm. (2.25) 4 The is a very technical result, the so-called Staeckel conditions, which gives necessary and sufficient conditions for separability in orthogonal coordinate systems.
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