On the relations between several notions of symmetry for the second order linear differential equation

Abstract

There are several non-equivalent notions of infinitesimal symmetry in the literature of second order linear differential equations: Lie point symmetries, vertical (gauge) symmetries, operator symmetries, infinitesimal contact symmetries, and Lie--Bäcklund operators. We construct an explicit correspondence among the first three, We then describe the Lie algebra ${\mathcal{L}}_{\Omega }(\mathbb{U})$ of infinitesimal contact transformations of the contact system $\Omega =⟨dy-y^{\prime }dx⟩$, with coefficients in a differential field $\U$ of functions of $x$ and $y$. We obtain a canonical decomposition ${\mathcal{L}}_{\Omega }(\mathbb{U})={\prod }_{k\ge 0}{\mathcal{L}}_{\Omega }^k\mathbb{U}$ into $\mathbb{C}$-vector spaces, each parametrized by $\mathbb{U}$ (by $\mathbb{U}\oplus \mathbb{U}$ for $k=0$), and we compute the algebraic differential formulae for the Lie bracket in these coordinates. Applied to the symmetry problem, we prove that a contact vector field with generating function $W$ is a symmetry of if and only if $A^{2}W=aW+bAW$, where $A$ is the vector field in the jet space corresponding to the equation; equivalently, if and only if $W=F_1(u_1,u_2){\varphi }_1+F_2(u_1,u_2){\varphi }_2$ for arbitrary functions $F_1,F_2$ of the two first integrals of $A$ and a basis ${\varphi }_1,{\varphi }_2$ of solutions. The symmetry algebra is always parametrized by two arbitrary functions of two variables. It also shows that the decomposition of $\Lom(\U)$ never captures the whole symmetry algebra: for $a\ne 0$ the graded part reduces to the point symmetries, while for $y^{\prime \prime }=0$ it is an infinite dimensional but still \emph{proper} subspace, and in neither case is it a Lie subalgebra. Finally we make precise the transformation law $W\mapsto {\mu }^{-1}(W\circ \phi )$ for characteristics under a contact transformation with conformal factor $\mu$, which governs the transport of evolutionary representatives.