Mahomed, F. M.; Kara, A. H.; Leach, P. G. L. Lie and Noether counting theorems for one-dimensional systems. (English) Zbl 0783.34002 J. Math. Anal. Appl. 178, No. 1, 116-129 (1993). Summary: For a second-order equation \(E(t,q,\dot q,\ddot q)=0\) defined on a domain in the plane, Lie geometrically proved that the maximum dimension of its point symmetry algebra is eight. He showed that the maximum is attained for the simplest equation \(\ddot q=0\) and this was later shown to correspond to the Lie algebra \(\text{sl}(3,\mathbb{R})\). We present an algebraic proof of Lie’s “counting” theorem. We also prove a conjecture of Lie’s, viz., that the full Lie algebra of point symmetries of any second-order equation is a subalgebra of \(\text{sl}(3,\mathbb{R})\). Furthermore, we prove, the Noether “counting” theorem, that the maximum dimension of the Noether algebra of a particle Lagrangian is five and corresponds to \(A_{5,40}\). Then we show that a particle Lagrangian cannot admit a maximal four-dimensional Noether point symmetry algebra. Consequently we show that a particle Lagrangian admits the maximal \(r\in\{0,1,2,3,5\}\)-dimensional Noether point symmetry algebra. Cited in 19 Documents MSC: 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 17B66 Lie algebras of vector fields and related (super) algebras Keywords:Lie’s counting theorem; Noether counting theorem; second-order equation; point symmetry algebra; Lie algebra; Noether algebra PDFBibTeX XMLCite \textit{F. M. Mahomed} et al., J. Math. Anal. Appl. 178, No. 1, 116--129 (1993; Zbl 0783.34002) Full Text: DOI