2 L. D. Faddeev and O. A. Yakubovski˘ı

are described by the Hamiltonian equations

(4) ˙i q =

∂H

∂pi

, ˙i p = −

∂H

∂qi

, i = 1,2, . . . , n.

Here H = H(q1, . . . , qn; p1, . . . , pn) is the Hamiltonian function, qi

and pi are the generalized coordinates and momenta, and n is called

the number of degrees of freedom of the system. We recall that for a

conservative system, the Hamiltonian function H coincides with the

expression for the total energy of the system in the variables qi and pi.

We write the Hamiltonian function for a system of N material points

interacting pairwise:

(5) H =

N

i=1

pi 2

2mi

+

N

ij

Vij (xi − xj ) +

N

i=1

Vi(xi).

Here the Cartesian coordinates of the particles are taken as the gener-

alized coordinates q, the number of degrees of freedom of the system

is n = 3N, and Vij (xi − xj ) is the potential of the interaction of the

ith and jth particles. The dependence of Vij only on the difference

xi −xj is ensured by Newton’s third law. (Indeed, the force acting on

the ith particle due to the jth particle is Fij = −

∂Vij

∂xi

=

∂Vij

∂xj

= −Fji.)

The potentials Vi(xi) describe the interaction of the ith particle with

the external field. The first term in (5) is the kinetic energy of the

system of particles.

For any mechanical system all physical characteristics are func-

tions of the generalized coordinates and momenta. We introduce the

set A of real infinitely differentiable functions f(q1, . . . , qn; p1, . . . , pn),

which will be called observables.1 The set A of observables is obvi-

ously a linear space and forms a real algebra with the usual addition

and multiplication operations for functions. The real 2n-dimensional

space with elements (q1, . . . , qn; p1, . . . , pn) is called the phase space

and is denoted by M. Thus, the algebra of observables in classical

mechanics is the algebra of real-valued smooth functions defined on

the phase space M.

We shall introduce in the algebra of observables one more opera-

tion, which is connected with the evolution of the mechanical system.

1We

do not discuss the question of introducing a topology in the algebra of ob-

servables. Fortunately, most physical questions do not depend on this topology.