Some Remarks on Classical Lagrangian Symmetric Differential Expressions and their Composite Powers

In this paper, we prove some general results about Lagrangian symmetric ordinary differential expressions [·] of order n when the coefficients of [·] are sufficiently smooth. In particular, for a natural number j , we show that under increased smoothness conditions on the coefficients, the j th composite power j [·] of [·] is also Lagrangian symmetric. More generally, we show that if w is a symmetry factor for [·], then w is also a symmetry factor for j [·]. Several classical second-order examples are given to illustrate these results and their j th composite powers, for each j ∈ N, are given explicitly in Lagrangian symmetric form. AMS subject classification: Primary 34A30, 47B25, Secondary 47E05.