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Example sheet III: 16.2.09 Hamiltonian Dynamics S2 2008/09 Exercise 1 ( Liouville’s Theorem for freely falling particles ) A group of particles, all of the same mass, having initial heights q0 and vertical momenta p0 lie in the rectangle −a ≤ q0 ≤ a and −b ≤ p0 ≤ b. The particles fall freely in the earth’s gravitational ﬁeld for a time t. Find the area in phase space in which they lie and show by direct calculation that this area is still 4ab. Exercise 2 ( Modiﬁed Hamilton’s principle ) Consider variations of the functional t2 I= ˙ ˙ dtf (q, q, p, p) t1 in phase space with f (q, q, p, p) = pq − H(q, p, t), such that the variations vanish at the ˙ ˙ ˙ end points. Show that the stationary points deﬁne Hamilton’s equations if independent variations in p and q are assumed. Why does the condition of vanishing variation for p at the end points not matter? Exercise 3 ( Example for generating function approach ) Given the Hamiltonian 1 k H = p2 q 4 + 2 , 2 2q and the generating function √ Q F1 (q, Q) = − mk . q 1. Find the transformations Q = Q(q, p) , P = P (q, p) , 2. What is the new Hamiltonian function K(Q, P )? 3. Solve the problem in terms of the new variables Q, P . Exercise 4 ( Revision: Poisson bracket ) Verify the algebraic properties of Poisson brackets, namely [u, v] = −[v, u] , [au + bv, w] = a[u, w] + b[v, w] a, b constant , [uv, w] = u[v, w] + [u, w]v , and [u, [v, w]] + [v, [w, u]] + [w, [u, v]] = 0 Jacobi identity , where u = u(q, p, t), v = v(q, p, t) and w = w(q, p, t). Exercise 5 ( Criteria for canonical transformations ) q, p are canonical variables. The transformation √ Q = ln(1 + q cos p) , √ √ P = 2(1 + q cos p) q sin p , deﬁnes new variables Q, P . Show that 1. the transformation is canonical, 2. the transformation is generated by F3 (p, Q) = −(eQ − 1)2 tan p . Exercise 6 ( Evaluation of standard commutators ) A particle with mass m, position r and momentum p has angular momentum L = r × p. Evaluate [ri , Lj ], [pi , Lj ], [Li , Lj ] and [L2 , Li ]. Suppose now that the particle has charge e and moves in a background ﬁeld B and so the Lagrangian is 1 ˙ ˙ L = mr2 + er · A , 2 with B = × A. Show that e ˙ ˙ [ri , rj ] = ijk Bk . m2 Exercise 7* ( Revision: Generating functions ) A contact transformation (i.e. a family of canonical transformations with time as a pa- rameter) transforms the set of generalised coordinates and momenta (q, p) to a set of new generalised coordinates and momenta (Q, P ), qi → Qi (q, p, t) , pi → Pi (q, p, t). Such a transformation is canonical if there exists a new Hamiltonian K(Q, P , t) such that the form of Hamilton’s equations of motion are unchanged, i.e. ˙ ∂K ˙ ∂K Pi = − and Qi = + . ∂Qi ∂Pi The original Hamiltonian H is related to the new Hamiltonian K through the equations ˙ ˙ H(q, p, t) = p · q − L(q, q, t) , ˙ ¯ ˙ K(Q, P , t) = P · Q − L(Q, Q, t) , with ˙ ¯ ˙ dF1 (q, Q, t) L(q, q, t) = L(Q, Q, t) + . dt Show that this can be satisﬁed if the generating function F1 (q, Q, t) is such that ∂F1 ∂F1 ∂F1 pi = , Pi = − , and K = H + . ∂qi ∂Qi ∂t Find the corresponding conditions in the case where the generating functions take the forms F2 = F2 (q, P , t) , F3 = F3 (p, Q, t) and F4 = F4 (p, P , t) .

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posted: | 9/9/2009 |

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