文档介绍：Quantum Field Theory for Mathematicians:
Hamiltonian Mechanics and Symplectic
Geometry
We’ll begin with a quick review of classical mechanics, expressed in the
language of modern geometry. There are two general formalisms used in classical
mechanics to derive the classical equations of motion: the Hamiltonian and
Lagrangian. Both formalisms lead to the same equations of motion in the cases
where they both apply, but they provide rather different points of view on both
classical mechanics and the subject we will turn to next, quantum mechanics.
For more detailed information about this subject, some good references are
[1], [2], [3] and [4].
1 Hamiltonian Mechanics and Symplectic Ge-
ometry
The standard example of classical mechanics in its Hamiltonian form deals with
a single particle moving in space (R3). The state of the system at a given time t
is determined by six numbers, the coordinates of the position (q1, q2, q3) and the
6
momentum (p1, p2, p3). The space R of positions and momenta is called “phase
space.” The time evolution of the system is determined by a single function of
these six variables called the Hamiltonian and denoted H. For the case of a
particle of mass m moving in a potential V (q1, q2, q3),
1
H = (p2 + p2 + p2) + V (q , q , q )
2m 1 2 3 1 2 3
The time evolution of the state of the system is given by the solution of the
fo