§ 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