Lie point symmetry analysis of a second order differential equation with singularity. (English) Zbl 1377.34045
A primary motivation of this work is to analyze the Lie point symmetries of the second order differential equation with cubic singularity
\[
\ddot{x}(t)+p(t)x(t)=qx(t)^{-3}+g(t)x(t)^m,
\]
where \(q, m\) are constants and \(p(\cdot), g(\cdot)\) are arbitrary functions of \(t\). Such equation arises in several applications coming from classical and quantum mechanics and geometry.
The first objective is to identify a class of equations above invariant under a \(1\)-dimensional subgroup of the symmetry group of the familiar Ermakov-Pinney equation. Under this condition, the given equation can be reduced to an equivalent autonomous equation by means of a canonical transformation. A second objective is to explore a generalization of the associated Ermakov-Lewis invariant.
The first objective is to identify a class of equations above invariant under a \(1\)-dimensional subgroup of the symmetry group of the familiar Ermakov-Pinney equation. Under this condition, the given equation can be reduced to an equivalent autonomous equation by means of a canonical transformation. A second objective is to explore a generalization of the associated Ermakov-Lewis invariant.
Reviewer: Mircea Crâşmăreanu (Iaşi)
MSC:
| 34C14 | Symmetries, invariants of ordinary differential equations |
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
Keywords:
Lie point symmetry; nonlinear superposition; Ermakov-Pinney equation; Ermakov-Lewis invariant; second order Kummer-Schwarz equation; \(\mathrm{sl}(2, \mathbb{R})\) invariant equationsReferences:
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