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.
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