confusion about CTs

Confusion about CT's PhL 8.4.08 Suppose we have the HJE and we have some S that is not an explicit function of time. Then we know that K = H. [ these notes relate to Goldstein Chapter 8 on canonical transformations I call CT's ] 1. What exactly do we mean by "K = H" ? We know for example that this happens when we use S = ΣiqiPi. The CT equations in this case say p = P and q = Q, so this is a confusing case. Consider some other S like that shown page 245 bottom, where S = ΣiqiQi. The equations say p = Q and P = -q. I think "the meaning" of K = H is this: K(P,Q) = H(p,q) in value, if you evaluate K at P,Q and H at p,q. I do NOT think these two functions have the same functional form. So we are saying, f1(P(p,q),Q(p,q) = f2(p,q) so f1 and f2 are different functions, but are equal when evaluated at corresponding phase space points. So in our last example, we would have K(P,Q) = H(p,q) = K(-q, p) If we let -q = x and p = y, then we have K(-q, p) = K(x,y) = H(y,-x) Suppose H(x,y) = x + 5y. Then K(x,y) = H(y,-x) = y - 5x. So clearly H and K are "different functions" of their arguments. 2. Given H(p,q) and p and q and some specific S(q,P), how do you find the P and Q? The CT equations in this case tell us that pi = ∂S/∂qi and Qi = ∂S/∂Pi. So this at least tells us what Qi is, but what are the Pi? __________________________________________ Let's try an example: H(p,q) = ap2 + bq2 //harmonic oscillator S(q,P) = 5qP The CT equations say p = ∂S/∂q = 5P so we think P = p/5. Other says Q = ∂S/∂P = 5q so we conclude that P = p/5 Q = 5q K(P,Q) = H(p,q) = H(5P,Q/5) = a(5P)2 + b(Q/5)2 = 25aP2 + bQ2/25 Let's check Hamilton's equations. We have = - ∂H/∂q = 2bq and = ∂H/∂p = 2ap = - ∂K/∂Q = 2bQ/25 and = ∂K/∂P = 50aP Are these consistent? = 2bQ/25 => /5 = 2b(5q)/25 => = 2bq yes = 50aP => 5 = 50 a(p/5) => = 2ap yes ___________________________________________ So I was saying Qi = ∂S/∂Pi. So this at least tells us what Qi is, but what are the Pi? If we write the other equation which is pi = ∂S(qi,Pi)/∂qi = G(qi,Pi) , we have a system of n equations i = 1..n which we can hopefully solved for the n variables Pi! Conclusion: the two CT equations tell you what P and Q will be! ______________________________________________ Another example, this time with W, as per page 286 G: assume a separable form such that W(q,P) = Σi Wi(qi, Pi) = W1(q1, P1) + W2(q2, P2) + ... Also assume that H(p,q) is cyclic in all coordinates except q1. From Hamilton's equations, we know that i = ∂H/∂qi = 0 constants for i = 2...n. so we know that pi = αi constant for i=2..n. Now that we know the pi are constants, we can use the CT equation pi = ∂W/∂qi = ∂Wi/∂qi = αi and conclude that we have Wi = αiqi for i=2..n. We then have W = W1(q1, P1) + Σiαiqi. Now, what can we say about Pi and Qi corresponding to this W and H? We have already used one of the CT equations to find that pi = αi fir i=2..n . The other equation says Qi = ∂W/∂Pi = ∂Wi/∂Pi. This tells us then that Q1= ∂W1/∂P1 and Qi = 0 for i = 2...n. But what are the Pi ? We know K(P,Q) = K(P,Q1) since all the other Q's are 0. This K is cyclic in all the other Qi and so we should have Pi = γi = constants for i=2..n. Can I show that γi = αi as suggested by the page 286 footnote?? The answer is no, you cannot show they are equal. But we know we can choose each set arbitrarily. So if we were to select γi = αi, then indeed we would have Pi = pi for i=2..n. In this case we could write W = W1(q1, P1) + Σi=2..nPiqi. and in this case we recognize the sum as the identify CT for those coordinates. So we would expect then that Pi = pi (which are then both αi), and Qi = qi . But the qi don't even show up in our H, so we can select them 0, and we have Qi= qi = 0, and indeed we have the identify situation for the n=2..n coordinates. 3. Clarification of the notion of "separable" in HJ theory. Discussion starts on page 284 and I failed the first time to understand it. We know that the most general form of an F2 W is W(q,P). We could of course imagine that there are also constants α and these have nothing to do with our previous uses of αi. Then we might have W(q,P, α). Now let's first restrict to the case where we select W's which don't depend on P, so we have W = W(q, α) . Now let's second assume that we can write W = Σi Wi(qi, α). That is, each Wi is a function only of coordinate qi. This is certainly a notion of "separation of variables" but in a summation sense. Now let's jump ahead to the 3D Kepler solution which starts on page 299. Since energy is conserved, we have our familiar constant α1 = E and we shall regard this in fact as α1 of our set α. So we have not said one of our constants is energy and we name that to be α1. In the Kepler problem, we then write W(r,θ,φ, α) = Wr(r,α) + Wθ(θ,α) + Wφ(φ,α) By waving of the arms properly, we are able then to come up with three different equations which have this form fφ(φ, ∂Wφ/∂φ, α) = 0; // see 9-63a which involves only αφ, one of our α fθ(θ, ∂Wθ/∂θ, α) = 0; // see 9-63b which involves αφ and αθ fr(r, ∂Wr/∂r, α) = 0; // see 9-63c which involves αθ and α1 The various αi constants just appear when you go through the normal separation of variables procedure with a multi-variable differential equation such as H = E for central potential. Traditionally we are doing perhaps QM and we have Hψ = Eψ and we try ψ as the product of three functions, each in its own variable, and we call that "separation of variables". But our HJ equation is different, although we still write it as H = E. The HJE says H(qi, pi= ∂Wi/∂qi, α) = E = α1. In the case of 3D Kepler, we find that we are able to "separate" the HJE into the three equations shown above. These equations are violently different from each other, but we can combine the above three into one by saying: fi(qi, ∂Wi/∂qi, α) = 0 i = r,θ,φ Although we started with a Hamiltonian H, these three equations are not themselves Hamiltonians, they are just functions. Suppose we take our three fi equations, whatever they are, and define three new functions Hi (having nothing to do with Hamiltonians! ) Hi(qi, ∂Wi/∂qi, α) ≡ fi(qi, ∂Wi/∂i, α) + αi Then of course we end up with these three equations Hi(qi, ∂Wi/∂qi, α) ≡ αi and this is what we see on page 284 equation 9-23. Notice these equations are typically not linear, but they are "first order" since only the first derivative of Wi appears, and they are "ordinary" because there is only one variable qi . Usually we get (∂Wi/∂qi)2 in the equation and we just solve it by taking a square root of all the other stuff. In the Kepler example, we could interpret the three equations as follows: Hφ z (pφ = ∂Wφ/∂φ) = αφ = m Hθ 2 (pφ = m, pθ = ∂Wθ/∂θ) = αθ2 = L2 Hr H ( pr = ∂Wr/∂r, L2) = E They are "like" Hamiltonians in that they all represent constants of the motion. So, when G writes these equations as Hi(qi, ∂Wi/∂qi, α) ≡ αi, he is just emphasizing that each of these equations is going to be associated with some conserved quantity, and he writes that on the right hand side as the corresponding αi. They are analogous to H = α1. We don't have H = ΣiHi. So, this then is the meaning of separability in HJ theory and this finally explains 9-23 page 284.