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Lie and Noether counting theorems for one-dimensional systems. (English) Zbl 0783.34002
Summary: For a second-order equation $E\left(t,q,\stackrel{˙}{q},\stackrel{¨}{q}\right)=0$ 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 $\stackrel{¨}{q}=0$ and this was later shown to correspond to the Lie algebra $\text{sl}\left(3,ℝ\right)$. 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 $\text{sl}\left(3,ℝ\right)$. 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 $r\in \left\{0,1,2,3,5\right\}$-dimensional Noether point symmetry algebra.
##### MSC:
 34A25 Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.) 34C20 Transformation and reduction of ODE and systems, normal forms 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems 17B66 Lie algebras of vector fields and related (super)algebras