§ 1. The algebra of observables in classical
mechanics
We consider the simplest problem in classical mechanics: the problem
of the motion of a material point (a particle) with mass m in a force
field V (x), where x(x1, x2, x3) is the radius vector of the particle. The
force acting on the particle is
F = −grad V = −
∂V
∂x
.
The basic physical characteristics of the particle are its coordi-
nates x1, x2, x3 and the projections of the velocity vector v(v1, v2, v3).
All the remaining characteristics are functions of x and v; for exam-
ple, the momentum p = mv, the angular momentum l = x × p =
mx × v, and the energy E =
mv2/2
+ V (x).
The equations of motion of a material point in the Newton form
are
(1) m
dv
dt
= −
∂V
∂x
,
dx
dt
= v.
It will be convenient below to use the momentum p in place of the
velocity v as a basic variable. In the new variables the equations of
motion are written as follows:
(2)
dp
dt
= −
∂V
∂x
,
dx
dt
=
p
m
.
Noting that
p
m
=
∂H
∂p
and
∂V
∂x
=
∂H
∂x
, where H =
p2
2m
+ V (x) is the
Hamiltonian function for a particle in a potential field, we arrive at
the equations in the Hamiltonian form
(3)
dx
dt
=
∂H
∂p
,
dp
dt
= −
∂H
∂x
.
It is known from a course in theoretical mechanics that a broad
class of mechanical systems, and conservative systems in particular,
1
http://dx.doi.org/10.1090/stml/047/01
