Lie and Noether Counting Theorems for One-Dimensional Systems

For a second-order equation E(t, q, q̇, q̈) = 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 q̈ = 0 and this was later shown to correspond to the Lie algebra sl(3, 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 sl(3, R). Furthermore, we prove, the Noether "counting" theorem, that the maximum dimension of the Noether algebra of a particle Lagrangian is live and corresponds to A5,40. 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 ∈ {0, 1, 2, 3, 5}.dimensional Noether point symmetry algebra.