Download PDF by L. A. PARS (President of Jesus College, Cambridge): A Treatise on ANALYTICAL DYNAMICS

By L. A. PARS (President of Jesus College, Cambridge)

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If the field is a uniform field -mg in the direction Oz the potential energy is V = mgz, and if the field is an attraction T (r) to the origin 0, where r = V(x2 + y2 + z2), we have V= J r 9,($) Particular cases are (i) the isotropic oscillator, where T(r) = mn2r, V = and (ii) the Newtonian attraction, where q(r) = um/r2, V = -,um/r. In all these cases there is a constant of integration which can be given any convenient value; in practice we are only interested in the change in V when the system moves from one configura2mn2r2, tion to another.

The force exerted on the particle by the external agent is a force of constraint. The classical example is that in which the system is a simple rigid body, one point of which is held and moved in a given way. (iii) The surface of a rigid body belonging to the system slides on a smooth surface which is fixed or which has an inexorable motion. The action of the smooth surface on the body is a force of constraint. (iv) The surface of a rigid body belonging to the system rolls without sliding on a surface which is fixed or which has an inexorable motion.

The scope of the definition could have been made wider in various ways. (i) We have assumed that all the forces acting on the particles of the system are either given forces or forces of constraint. We could have adopted a wider definition to include 26 A TREATISE ON ANALYTICAL DYNAMICS systems for which this is not true. A conspicuous example is provided by a particle sliding on an imperfectly rough surface. The reaction of the surface on the particle is determined by the rule that the tangential component is in a direction directly opposite to the direction of motion, and its magnitude is It times the normal component, where It is a physical constant depending on the nature of the surface at the point.

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