Student Seminar: Classical and Quantum Integrable Systems

7. Homework exercises 95
7.1 Seminar 1 95
7.2 Seminar 2 96
7.3 Seminar 3 97
7.4 Seminar 4 98
7.5 Seminar 5 100
7.6 Seminar 6 102
7.7 Seminar 7 105
7.8 Seminar 8 107
1. Liouville Theorem
1.1 Dynamical systems of classical mechanics
To motivate the basic notions of the theory of Hamiltonian dynamical systems consider
a simple example.
Let a point particle with mass m move in a potential U(q), where q = (q 1 , . . . q n )
is a vector of n-dimensional space. The motion of the particle is described by the
Newton equations
m¨q i = − ∂U
∂q i
Introduce the momentum p = (p 1 , . . . , p n ), where p i = m ˙q i and introduce the energy
which is also know as the Hamiltonian of the system
H = 1
2m p2 + U(q) .
Energy is a conserved quantity, i.e. it does not depend on time,
dH
dt = 1 m p iṗ i + ˙q i ∂U
∂q i = 1 m m2 ˙q i¨q i + ˙q i ∂U
∂q i = 0
due to the Newton equations of motion.
Having the Hamiltonian the Newton equations can be rewritten in the form
˙q j = ∂H
∂p j
,
ṗ j = − ∂H
∂q j .
These are the fundamental Hamiltonian equations of motion. Their importance lies
in the fact that they are valid for arbitrary dependence of H ≡ H(p, q) on the
dynamical variables p and q.
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