§ 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

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