Infinitesimal symmetry transformations. II. Some one‐dimensional nonlinear systems

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Abstract
The converse problem of similarity analysis is discussed for the infinitesimal symmetry transformations of ordinary second‐order differential equations which are nonlinear in ẋ (and may be linear or nonlinear in x). A natural classification of the problem arises, according to the highest order N of nonlinearity in ẋ. The completely general maximal Lie algebra is obtained for the case N≤3. In the case N≥4 one has, besides the system of differential equations for the infinitesimal generators, an extra set of anholonomic constraints, which operates as a symmetry‐breaking mechanism producing a strong reduction in the number of surviving parameters. Miscellaneous examples are given, which illustrate some features of similarity analysis of nonlinear systems. The infinitesimal point transformation symmetries of the Van der Pol oscillator are also briefly discussed.

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