The Property of Covariance in Hori’s Noncanonical Perturbation Theory

Abstract

The present paper is strongly based on the work of Hori (Hori, 1966, 1971). In his 1966 paper Hori introduced the idea of Lie series in the field of canonical perturbation theory, in his 1971 article he applied the same idea to noncanonical autonomous systems of differential equations.

While the canonical invariance of Hori’s 1966 theory was already mentioned by Hori in the 1966 publication, it seems that the analogous property in the noncanonical case, the covariance with respect to arbitrary transformations, has not yet been recognized.

Let us briefly characterize the property of covariance and its consequences.

Consider some perturbed problem, given in terms of some set of variables. Consider the attached perturbation equations and assume them to be solved. Consider next the same problem, now in terms of some different set of variables and the corresponding perturbation equations. The property of covariance allows us to establish the relation between the solutions of the two sets of perturbation equations.

It is possible to base an integration procedure of the perturbation equations on the property of covariance. In order to emphasize the difference between the usual integration procedure of the perturbation equations and the one presented in this paper, let us recall the usual approach:

A perturbation problem is given in terms of some variables, say cartesian coordinates. Making use of the fact that the unperturbed problem is assumed to be solvable, a new set of variables, the elements, are introduced and the perturbed problem is formulated in terms of these variables. By solving the attached perturbation equations a near identical transformation is performed. The very final step is the solution of the remaining, the so called longperiodic, system:

Thus: first we perform the transformation to elements and afterwards the near identical transformation is established.

It is natural to ask if it is not possible to perform first a near identical transformation.

If we write down the perturbation equations in terms of the original variables they are, in general, fairly complicated and not easy to handle.

By virtue of the property of covariance, however, we are not obliged to solve these perturbation equations in terms of the original variables. We can choose a more convenient set of variables, for instance switch to elements. The perturbation equations expressed in terms of the elements are treated in the usual manner (elimination of short periodic terms).

This procedure generates a near identical transformation and a remaining long-periodic system of differential equations, expressed in terms of ‘slightly-changed-elements’.

The property of covariance permits one to formulate an equivalent near identical transformation and an equivalent remaining system of differential equations in terms of ‘slightly-changed-original-variables’.

This very system of differential equations must still be solved and for this purpose it may be useful to transform it to elements. The resulting system of differential equations is obviously identical with the remaining longperiodic system of the usual approach.

Thus, first we perform the near identical transformation, afterwards the transformation to elements is applied.

Therefore, in the present approach the two basic transformations are interchanged. What is the use of this interchange?

It is known from the canonical case, that the interchange of the two basic steps allows one to remove certain difficulties which are produced by the fact that the set of elements used are singular. The author expects that, by means of the present approach, similar difficulties in the noncanonical case become removable.

More details are presented in the author’s paper: The Transformational Behaviour of Perturbation Theories.