Lectures on String Theory

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Again we conclude from here that the Hamiltonian H is a time-conserved quantity
dH
dt
= {H, H} = 0 .
Thus, the motion of the system takes place on the subvariety of phase space defined
by H = E constant.
In the case under consideration the matrix J is non-degenerate so that there
exist the inverse
J −1 = −J
which defines a skew-symmetric bilinear form ω on phase space
ω(x, y) = (x, J −1 y) .
In the coordinates we consider it can be written in the form
ω = ∑ j
dp j ∧ dq j .
This form is closed, i.e. dω = 0.
A non-degenerate closed two-form is called symplectic and a manifold endowed
with such a form is called a symplectic manifold. Thus, the phase space we consider
is the symplectic manifold.
Imagine we make a change of variables y j = f j (x k ). Then
ẏ j = ∂yj
}{{} ∂x k
A j k
ẋ k = A j k J km ∇ x mH = A j km ∂yp
kJ ∂x m ∇y pH
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