By Goldstein Herbert

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We use the notation τb (t0 ) to indicate that this is the proper time at which the bottom of the rocket emits the light that arrives at the observer’s origin at the observer’s time t0 . At this proper time, from (2), the position of the bottom of the rocket in the observer’s reference frame was zb (τb (t0 )) = −γL0 /2 + vγτb (t0 ) 1/2 1+β 1−β = −γL0 /2 + vγ t0 − L0 2c (3) Similarly, for the top of the rocket we have 1+β 1−β τt (t0 ) = 1/2 t0 + L0 2c and zt (τt (t0 )) = γL0 /2 + vγ 1+β 1−β 1/2 t0 + L0 2c (4) Subtracting (3) from (4), we have the length for the rocket computed by the observer from his observations at time t0 in his reference frame: L(t0 ) = γ(1 + β)L0 = 1+β 1−β 1/2 L0 .

The final form of the generating function is then ψ(x, Q1 ) = F (x, Q1 , py , Q2 ) = − α 1 x + py Q2 + 2 α α 2 x2 1 1 − xpy + 2 p2y cot Q1 8 2 2α and its existence proves the canonicality of the transformation. e. B = B0 k, A= B0 2 − y ˆi + x ˆj . Then the Hamiltonian is 2 1 q p− A 2m c qB0 1 px + y = 2m 2c H(x, y, px , py ) = = 1 2m px + α2 y 2 2 + py − 2 + py − qB0 x 2c α2 x 2 2 2 where we put α2 = qB/c. In terms of the new variables, this is 1 α 2m α2 = P1 m = ω c P1 H(Q1 , Q2 , P1 , P2 ) = 2P1 cos Q1 2 + α 2P1 sin Q1 2 where ωc = qB/mc is the cyclotron frequency.

Q = (2) When applied to the case at hand, all four of these yield the same condition, namely 1 µ=− . 2iν For µ = ν = 1, which is the case Goldstein gives, these conditions are clearly 1 we see that not satisfied, so (1) is not canonical. But putting µ = 1, ν = − 2i equations (1) are canonical. 2 (a) For a one-dimensional system with the Hamiltonian p2 1 − 2, 2 2q H= show that there is a constant of the motion D= pq − Ht. 2 (b) As a generalization of part (a), for motion in a plane with the Hamiltonian H = |p|n − ar−n , where p is the vector of the momenta conjugate to the Cartesian coordinates, show that there is a constant of the motion D= p·r − Ht.

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