Student Seminar: Classical and Quantum Integrable Systems

or in the matrix form
ẏ = AJA t · ∇ y H .
The new equations for y are Hamiltonian if and only if
AJA t = J
and the new Hamiltonian is ˜H(y) = H(x(y)).
Transformation of the phase space which satisfies the condition
AJA t = J
is called canonical. In case A does not depend on x the set of all such matrices form
a Lie group known as the real symplectic group Sp(2n, R) . The term “symplectic
group” was introduced by Herman Weyl. The geometry of the phase space which
is invariant under the action of the symplectic group is called symplectic geometry.
Symplectic (or canonical) transformations do not change the symplectic form ω:
ω(Ax, Ay) = −(Ax, JAy) = −(x, A t JAy) = −(x, Jy) = ω(x, y) .
In the case we considered the phase space was Euclidean: M = R 2n . This is not
always so. The generic situation is that the phase space is a manifold. Consideration
of systems with general phase spaces is very important for understanding the
structure of the Hamiltonian dynamics.
1.2 Harmonic oscillator
Historically it is proved to be difficult to find a dynamical system such that the
Hamiltonian equations could be solved exactly. However, there is a general framework
where the explicit solutions of the Hamiltonian equations can be constructed. This
construction involves
• solving a finite number of algebraic equations
• computing finite number of integrals.
If this is the way to find a solution then one says it is obtained by quadratures.
The dynamical systems which can be solved by quadratures constitute a special
class which is known as the Liouville integrable systems because they satisfy the
requirements of the famous Liouville theorem. The Liouville theorem essentially
states that if for a dynamical system defined on the phase space of dimension 2n one
finds n independent functions F i which Poisson commute with each other: {F i , F j } =
0 then this system van be solved by quadratures.
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