Lectures on String Theory

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Appendices
A. 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.
The last two equations can be rewritten in terms of the single equation. Introduce
two 2n-dimensional vectors
( )
(
∂H
p
x = , ∇H =
q
)
∂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 .