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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,q ˙,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 q ¨=0 and this was later shown to correspond to the Lie algebra sl(3,). 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,). 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{0,1,2,3,5}-dimensional Noether point symmetry algebra.
34A25Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)
34C20Transformation and reduction of ODE and systems, normal forms
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
17B66Lie algebras of vector fields and related (super)algebras