Student Seminar: Classical and Quantum Integrable Systems

The last two equations can be rewritten in terms of the single equation. Introduce
two 2n-dimensional vectors
( ) p
x = , ∇H =
q
(
∂H
)
∂p j
∂H
∂q j
and 2n × 2n matrix J:
J =
( ) 0 −I
I 0
Then the Hamiltonian equations can be written in the form
ẋ = J · ∇H , or J · ẋ = −∇H .
In this form the Hamiltonian equations were written for the first time by Lagrange
in 1808.
Vector x = (x 1 , . . . , x 2n ) defines a state of a system in classical mechanics. The
set of all these vectors form a phase space M = {x} of the system which in the present
case is just the 2n-dimensional Euclidean space with the metric (x, y) = ∑ 2n
i=1 xi y i .
The matrix J serves to define the so-called Poisson brackets on the space F(M)
of differentiable functions on M:
{F, G}(x) = (∇F, J∇G) = J ij ∂ i F ∂ j G =
n∑
j=1
( ∂F
∂p j
∂G
∂q j − ∂F
∂q j ∂G
∂p j
)
.
Problem. Check that the Poisson bracket satisfies the following conditions
{F, G} = −{G, F } ,
{F, {G, H}} + {G, {H, F }} + {H, {F, G}} = 0
for arbitrary functions F, G, H.
Thus, the Poisson bracket introduces on F(M) the structure of an infinitedimensional
Lie algebra. The bracket also satisfies the Leibnitz rule
{F, GH} = {F, G}H + G{F, H}
and, therefore, it is completely determined by its values on the basis elements x i :
{x j , x k } = J jk
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