ABSTRACT
This chapter deals with the most important topic of this book, i.e., the nontrivial symmetries that a given ordinary differential equation (ode) may admit. Let two coordinate sets of the plane be defined with the additional assumption that in either of them one coordinate variable is dependent on the other. Roughly speaking a symmetry of an ode is a diffeomorphism connecting these coordinates for which this ode is an invariant. In the literature on symmetries these transformations are often called point transformations or variable transformations in order to distinguish them from more general transformations also involving the first derivative. This notation will frequently be used from now on whenever symmetries of an ode are the main topic. It is obvious that the entirety of symmetries of any given ode forms a
group. The term symmetry group of a differential equation is applied to the largest group of transformations sharing this property. The Lie algebra of its infinitesimal generators forms the corresponding symmetry algebra. If a variable transformation is applied to a given differential equation, the symmetry group of the transformed equation is similar to the original one according to Definition 3.2. The equivalence class to which the symmetry group of a particular ode belongs is called its symmetry type. Consequently, all equations contained in an equivalence class have the same symmetry type. The reverse is not true. As a consequence, the entirety of all differential equations allowing the same type of symmetry group is the union of equivalence classes. Krause and Michel [96] called it the stratum of ode’s corresponding to a symmetry type. In this book it will be called the symmetry class. The symmetries of a differential equation occur in different connections. On
the one hand, there is the classification problem. Its aim is to determine all possible symmetry types for a family of ode’s, e. g. ode’s of a fixed order. The starting point for this approach is the listing of groups given in Section 3.4; its differential invariants determine the general form of an ode that may be invariant under the respective group. On the other hand, if any particular ode is given, its symmetry type has to be determined if it is to be applied for finding its solutions. The symmetry problem for equations of order one is significantly different
from that of equations of order two or higher. The main difference is the fact that for first order equations in general there is no algorithm available for determining any symmetry generator; only heuristics or insight into the
problem from which the equation originates may allow finding them. What makes the problem especially difficult is the fact that any first order ode allows infinitely many symmetries as discussed at the beginning of Section 5.2. Contrary to this, for equations of order two or higher, the symmetry type may always be identified algorithmically, and there is a well-defined procedure for transforming it to a canonical form. The key for this is the Janet basis for the determining system and the theorems derived in Section 3.5. In the subsequent Section 5.1, general properties of the behavior of differen-
tial equations under a change of variables are discussed and various concepts related to its symmetries are defined. Section 5.2 discusses the symmetry structure of first order equations. The most important field of application for Lie himself was equations of second order; they are the subject of Section 5.3. In Section 5.4 a complete discussion of equations of order three is given following a similar approach to the second order equations. Finally in Section 5.5 the symmetries of linear equations of any order are discussed.
