1 Graded problems. PHY 5246: Theoretical Dynamics, Fall November 23 rd, 2015 Assignment # 12, Solutions. Problem 1

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1 PHY 546: Theoretical Dynaics, Fall 05 Noveber 3 rd, 05 Assignent #, Solutions Graded probles Proble.a) Given the -diensional syste we want to show that is a constant of the otion. Indeed,.b) dd dt Now we consider a plane otion with H p q, () D pq Ht () [D, H + D [ pq t Ht, H + D t [ pq, H H [p, Hq + p[q, H H [ p, q + [q, q p p H ( ) q + ( p q p p 3 ) q q + p p + q 0. and we want to show there is a constant of the otion (3) H p n ar n, (4) D n Ht, (5)

2 where r (x, y, z) and p is the vector of the conjugate oenta to (x, y, z). dd [D, H + H [ dt t n Ht, H H (6) [ [ n, H xpx + yp y + zp z H, p n a H n r [ [ n r n, p n a, H r [ n p i x i, ( [ p n j) n/ a p i x i, n ( H i j i j x j )n/ n p k n p k( p j) n/ + a nx k n ( H j j x j )n/+.c) k ( p ) n + ar n ( p ) n ar n. Consider the canonical transforation Q λq, p λp but with the tie dilation t λ t. Clearly the canonical for of Hailton equations is not preserved if t scale as in t λ t, but we can check that the analytic for of the equations (in ters of (q, p) in one case and (Q, P ) in the other) is the sae. Indeed, q [q, H [q, p [q, p p, ṗ [p, H q [p, p q k [ p, q Ipleent the transforation we can then derive that q dq dt λ ( q ) 3 q. 3 dq dt dq dt λ dt dt dq λ dt λ λ dq (7) dt dq dt λ q λ p λ λp P Q P, ṗ dp dt λdp dt λdp dt dt dt λdp dt λ λ 3 dp dt dp dt λ ṗ 3 λ 3 q 3 Q P 3 Q, 3 and prove that the for of Hailton s equations is preserved. Proble.a) Given the Hailtonian H ( q + p q 4 ), (8)

3 the canonical equations of otion are q [q, H [ q, ( ) q + p q 4 (9) [q, p q 4 p[q, pq4 q 4 p, [ ṗ [p, H p, ( ) q + p q 4 [p, q + [p, p q 4 ( ) q 3 p 4q 3 q 3 p q 3..b) The canonical transforation that will ake H look like the the Hailtonian of a haronic oscillator of position Q and conjugate oentu P is { P { q q Q pq P p QP (0) This transforation takes The generating function is H H (P + Q ). () F q Q, () such that { F P q P q Q F p P q Q p Q QP. (3) The canonical equations in the (Q, P ) variables are { Q H P, P P. (4) H Q Q

4 which, using Q Q(q, p) and P P (q, p) as given in Eq. (0), becoe Q P (ṗq + pq q) q ṗ ( ) q q pq q (5) Proble 3 3.a) We can prove that the transforation P Q q q pq q pq 4 as before ṗ q ( q pq pq4 ) q 3 p q 3 as before. Q p + iaq, P p iaq ia, (6) is canonical either by showing that it preserves the for of Hailton s equations of otion, or by verifying that the Jacobian atrix M of the change of variables (q, p) (Q, P ) satisfy the syplectic condition J MJM T, or by verifying that the fundaental Poisson brackets are invariant under such transforation. The last two proofs are very siple. Indeed, the atrix M is given by, ( ) ia and one can easily verify that ( ia MJM T ia M ) ( 0 0 On the other hand, it is also easy to show that { {Q, P } p + iaq, p iaq } ia ia ) ( ia ia, (7) ) ( 0 0 ) J. (8) {p, q} + {q, p} {q, p} (9) such that the for of the fundaental Poisson brackets is preserved. In order to prove that the canonical for of the equations of otion is preserved we need to specify the Hailtonian. Fro part.b) we know that the syste is a one-diensional haronic oscillator. Therefore, in ters of (q, p) variables the Hailtonian is H p + k q, (0)

5 and the equations of otion are H ṗ kq, () q H p q p, which, upon further derivation with respect to tie, can be cast in the for, q + ω q 0 q(t) D cos(ωt + δ), () ṗ kq p(t) Dk sin(ωt + δ), ω where we can recognize the failiar solution for the one-diensional haronic oscillator in ters of two arbitrary constants (D and δ) that can be deterined fro the initial conditions q(t 0) q 0 and p(t 0) p 0. In order to find the for of Hailton s equations in ters of the new variables (Q, P ), we need to find the transfored hailtonian, H (Q, P ), which is obtained fro H(q, p) by replacing q q(q, P ) and p p(q, P ), where, through soe siple algebra, one see that, q Q iap ia Choosing a ω in Eq. (3), where ω k and p Q + iap, we can write. (3) H p + ω q In ters of Q and P variables, Hailton s equations are now, iωqp H, (4) H Q P iωp, (5) H P Q iωq, which, upon further derivation with respect to tie, can be cast in the for of the sae for of Eq. (). 3.b) P + ω P 0, (6) Q + ω Q 0, The for of the Hailtonian in ters of Q and P and the corresponding equations of otions have been found in.a). Here we want to find the solution of Eqs. (6) and show that it corresponds to the solution of Eqs. (), i.e. to the otion of a one-diensional haronic oscillator. It is indeed obvious fro the for of the equations, but, to be pedantic, let us write the solution of

6 Eqs. (6) (in ters of two arbitrary constants A and B which could be thought as A Q(t 0) and B P (t 0)) as Q(t) Ae iωt, (7) P (t) Be iωt. Substituting it into q(t) of Eq. (3) we get q(t) A ( ) ( A A ia eiωt Be iωt ia B cos(ωt) + i ia + B D cos δ cos(ωt) D sin δ sin(ωt) D cos(ωt + δ), which corresponds to q(t) in Eq. () if we identify ( ) ( ) A A D cos δ ia B and D sin δ i ia + B Inverting these relations to get A and B, ) sin(ωt) (8). (9) A iad(cos δ + i sin δ) iade iδ, (30) B D (cos δ i sin δ) D e iδ, and substituting the in Eq. (3) to obtain p(t) gives p(t) A eiωt + iabe iωt iadeiδ e iωt ia D e iδ e iωt (3) iad ( e i(ωt+δ) e i(ωt+δ)) ad sin(ωt + δ), which corresponds to p(t) of Eq. () for a ω k ω.