Lie and Noether counting theorems for one-dimensional systems. (English) Zbl 0783.34002
Summary: For a second-order equation 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 and this was later shown to correspond to the Lie algebra . 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 . Furthermore, we prove, the Noether “counting” theorem, that the maximum dimension of the Noether algebra of a particle Lagrangian is five and corresponds to . 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 -dimensional Noether point symmetry algebra.
|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|