Classical Mechanics

here xi = (xi1, . . . , xid) are coordinates of the i-th particle and ∂xi is the gradient (∂xi1 , . . . , ∂xid); d is the space dimension (i.e. d = 3, usually). The potential energy function will be supposed “smooth”, i.e. analytic except, possibly, when two positions coincide. The latter exception is necessary to include the important cases of gravitational attraction or, when dealing with electrically charged particles, of Coulomb interaction. A basic result is that if V is bounded below the equation (1.1) admits, given initial data X0 = X(0), Ẋ0 = Ẋ(0), a unique global solution t → X(t), t ∈ (−∞,∞); otherwise a solution can fail to be global if and only if, in a finite time, it reaches infinity or a singularity point (i.e. a configuration in which two or more particles occupy the same point: an event called a collision). In Eq. (1.1) −∂xiV (x1, . . . ,xn) is the force acting on the points. More general forces are often admitted. For instance velocity dependent friction forces: they are not considered here because of their phenomenological nature as models for microscopic phenomena which should also, in principle, be explained in terms of conservative forces (furthermore, even from a macroscopic viewpoint, they are rather incomplete models as they should be considered together with the important heat generation phenomena that accompany them). Another interesting example of forces not corresponding to a potential are certain velocity dependent forces like the Coriolis force (which however appears only in non inertial frames of reference) and the closely related Lorentz force (in electromagnetism): they could be easily accomodated in the upcoming Hamiltonian formulation of mechanics, see Appendix A2. The action principle states that an equivalent formulation of the equations (1.1) is that a motion t → X0(t) satisfying (1.1) during a time interval [t1, t2] and leading from X = X0(t1) to X 2 = X0(t2), renders stationary the action