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Lie and Noether counting theorems for one-dimensional systems. (English) Zbl 0783.34002

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.

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
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