# Example sheet III 16.2.09 Hamiltonian Dynamics S2 200809

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```					Example sheet III: 16.2.09                      Hamiltonian Dynamics           S2 2008/09

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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