Remarks on Lines of reversibility for Poincaré´s centre problem
where p, q are polynomials whose terms of lowest order are of degree at least two. A well-known sufficient condition, due to Poincare, for the origin to be a centre is that the system be reversible with respect to a line L, which passes through the origin, i.e. that the system be invariant under a reflection in the line L, and under a simultaneous reversal of the independent variable t. Thus system (1.1) is reversible with respect to the line x = 0 if and only if it is invariant under the transformation (x, y, t) → (−x, y,−t), i.e. if and only if q(−x, y) = q(x, y) and p(−x, y) = −p(x, y). Thus q contains only even powers of x and p only odd ones. Reversibility with respect to y = 0 is thus equivalent to q(x,−y) = −q(x, y), p(x,−y) = p(x, y), i.e. q contains only odd powers of y and p only even ones. As for a general line L one can apply a rotation which transforms L into the line x = 0 or y = 0 and a criterion for reversibility may then readily be attained [1]. Collins, by using tensor-calculus, derives in [2] a necessary and sufficient condition for the existence of such a line without involving its unknown equation.