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 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 |