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 LΩ(U) of infinitesimal contact transformations of the contact system Ω=⟨dy−y′dx⟩
Nearby in the stack
, with coefficients in a differential field
of functions of
x
and
y
. We obtain a canonical decomposition
LΩ(U)=∏k≥0LΩkU
into
C
-vector spaces, each parametrized by
U
(by
U⊕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
A2W=aW+bAW
, where
A
is the vector field in the jet space corresponding to the equation; equivalently, if and only if
W=F1(u1,u2)φ1+F2(u1,u2)φ2
for arbitrary functions
F1,F2
of the two first integrals of
A
and a basis
φ1,φ2
of solutions. The symmetry algebra is always parametrized by two arbitrary functions of two variables. It also shows that the decomposition of
()
never captures the whole symmetry algebra: for
a=0
the graded part reduces to the point symmetries, while for
y′′=0
it is an infinite dimensional but still proper subspace, and in neither case is it a Lie subalgebra. Finally we make precise the transformation law
W↦μ−1(W∘φ)
for characteristics under a contact transformation with conformal factor
μ
, which governs the transport of evolutionary representatives.
arXiv did not return neighbours for this paper. Try again in a bit.