(i) The Hamiltonian in classical mechanics (CM) is the total energy function exp

Question

(i) The Hamiltonian in classical mechanics (CM) is the total energy function expressed in phase-space coordinates as H H(q,p), where q and p are independent, N-dimensional vectors representing position and momentum respectively. A very useful quantity in CM is the Poission Bracket (PB). The PB between any two phase-space functions A(q, p) and B(q,p) where qi and p represent the ith component of the vectors q and p. The time-dependence of any function in CM, A(q, p), is expressed in terms of a PB with the Hamilto- nian, dA = {A,H) dt Given a 1-D system Hamiltonian p2 2m

Explanation / Answer

(a) Constant of motion is a quantity that is conserved throughout the motion i.e. it does not change with time. If A is a constant of motion, then dA/dt = 0. To show that H, Hx, Hy are constants of motion, we need to prove that (i) dH/dt = 0 (ii) dHx/dt = 0 (iii) dHy/dt = 0

(i) since dA/dt = {A,H} therefore dH/dt = {H,H} = 0

(ii) dHx/dt = {Hx,H} = {Hx,Hx+Hy} (since H= Hx+Hy) = {Hx,Hx} + { Hx,Hy} = 0 + (x)(0)-(px)(0) + (0)(py)-(0)(y) = 0 (using eqn 4)

(iii) dHx/dt = {Hx,H} = {Hx,Hx+Hy} ={Hy,Hx} + {Hy,Hy} = 0 (using eqn 4)

Hence we proved that H,Hx,Hy are constants of motion.

(b) F = p2/2

dF/dt = {F,H} = 0 (using eqn 4)

here F denotes the total energy of the system which is conserved.

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