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
I= ˙ ˙
dtf (q, q, p, p)
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
H = p2 q 4 + 2 ,
and the generating function
F1 (q, Q) = − mk .
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 ,
[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
1 ˙ ˙
L = mr2 + er · A ,
with B = × A. Show that
[ri , rj ] = ijk Bk .
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 = + .
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) ,
˙ ¯ ˙ dF1 (q, Q, t)
L(q, q, t) = L(Q, Q, t) + .
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) .