A New Look for Hamiltonian Dynamics

It seems a pity that Hamiltonian dynamics—contact transformations and so on—is regarded as a fearsome subject, too time-consuming to teach to most students; for it is the one branch of dynamics to point a way to new developments in this century. Moreover the basic ideas are extremely simple, but presented in an unfortunate way in all the text-books.

The Existence of Integrals of Dynamical Systems Linear in the Velocities

A dynamical system means here a system specified by generalised coordinates qα(α = 1, 2, …, n) and a Lagrangian L which is a quadratic polynomial in the generalised velocities, say(with a summation convention).

Inertia Invariants of a Set of Particles

The formal object of this note is the calculation of the principal moments of inertia of a set of particles at their mass centre, in terms of their mutual distances; he calculation brings in some identities which although simple may be in part novel.

Latent Roots of Tri-Diagonal Matrices

A considerable amount is known about the latent roots of matrices of the formin the case when each cross-product of non-diagonal elements, aici-1, is positive. One forms the sequence of polynomials fr(λ) = |Lr−λI| for r = 1, 2, … n, and observes thatthen it is easy to deduce that (i) the zeros of fn(λ) and fn_1(λ) interlace—that is, between two consecutive zeros of either polynomial lies precisely one zero of the other (ii) at the zeros of fn(λ) the values of fn-x(λ) are alternately positive and negative, (iii) all the zeros of fn(λ)— i.e. all the latent roots of Ln—are real and different.

The Electrostatic Energy of a Two-Dimensional System

The use of the complex variable z( = x + iy) and the complex potential W(= U + iV) for two-dimensional electrostatic systems is well known and the actual system in the (x, y) plane has an image system in the (U, V) plane. It does not seem to have been noticed previously that the electrostatic energy per unit length of the actual system is simply related to the area of the image domain in the (U, V) plane.

A Note on Gamma Functions

Various improvements in the formulawhich was discovered by Wallis in 1669, were studied by D. K. Kazarinoff in No. 40 of these Notes (December 1956).