Lectures on String Theory

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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.
Dynamical systems with symmetries
Let g(t) will be a one-parametric group of transformations of the phase space:
x → g(t)x. This group does not need to coincide with the one generated by the
Hamiltonian H. The action of this group is called Hamiltonian if there exists a
function C such that
d
dt g(t)x| t=0 = J · ∇C .
The flow of any function F under the one-parameter group generated by C is then
δF ≡ d dt F (g(t)x)| t=0 = ∇F · d
dt g(t)x| t=0 = (∇F, J · ∇C) = {F, C}
If we take F = H, we get
δH ≡ {H, C} .
Thus, if Ċ = {H, C} = 0, i.e. if C is an integral of motion, then it generates the
symmetry transformations which leave the Hamiltonian invariant. Infinitezimally,
the symmetry transformations are realized as
δF = {F, C} .
There could be several one-parametric groups which are one-parametric subgroups of
a non-abelian Lie G, the latter being the symmetry of the Hamiltonian. Accordingly,
there are the integrals of motion C i , i = 1, . . . , dim G. Since C i are integrals of
motion, from the Jacobi identity
{{C i , C j }, H} + {{H, C i }, C j } + {{C j , H}, C i } = 0
we conclude that {{C i , C j }, H} = 0, i.e. {C i , C j } is an integral of motion. If one can
chose the functions C i is such a way that they form the Lie algebra of G under the
Poisson bracket:
{C i , C j } = f k ijC k ,
then the corresponding action of G on the phase space is called Poisson. Here f k ij
are the structure constants of the Lie algebra of G.