Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

This phenomenon could be explained as follows: as soon as the tensors˜S j iklvanish, the system enjoys a projective connection (appendixof [Fe1995]); with such a connection, the tensors ˜P ji then transform accordingto a specific rule via tensorial rotation formulas and their general expressionmay be deduced from their expression at the identity 4 . We have checkedthis, but as we try to avoid the method of equivalence, details will not be reproducedhere. Similar observations appear in Hachtroudi [Ha1937].Even if the expressions (1.10) are more compact than the (equivalent)conditions in Theorem 1.7 (3), we prefer the complete expressions of Theorem1.7 (3), since they are more explicit and ready-made for checkingwhether a physically given nonlinear system is equivalent to a free particle.If the reader prefers compact expressions and “short” theorems, (s)hemay replace the conditions of Theorem 1.7 (3) by (1.10).Open problem 1.11. Characterize explicitly the linearizability of a Newtoniansystem in m 2 degrees of freedom, i.e. local equivalence to :m∑m(1.12) Y j XX = Gj 0(X) + Y l G j 1,l (X) + ∑Y l 1X Hj l 1(X).l=11.13. Organization, avertissement and acknowledgment. Section 2 is devotedto a thorough restitution of Lie’s original proof of the equivalence between(1) and (4) in Theorem 1.2. Section 3 is devoted to the formulationof combinatorial formulas yielding the general form of a system equivalentto Y j XX= 0, j = 1, . . ., m, under a local K-analytic point transformation(x, y j ) ↦→ (X, Y j ), for general m 2 ; the proof of the main technicalLemma 3.32 is exposed in Section 5. Section 4 is devoted to the final proofof the equivalence between (1) and (3) in Theorem 1.7, the equivalence between(1) and (2) being already proved by Fels [Fe1995].Some word about style and intentions. We wanted the proof of the equivalencebetween (1) and (3) be totally complete, every tiny detail being rigorouslyand patiently checked. This is why we decided to carefully detaileach intermediate computational step, seeking first the combinatorics of theformal calculations in the case m = 1 and devising then the underlying combinatoricsfor the case m 2. Actually, the size of differential expressionsis relatively impressive, as will become soon evident. Thus, no intermediatesymbolic computation will be hidden, hence essentially no checking workis left to the reader, as would have been the case if we did not have typed allthe computations.4 Similar rotation formulas are known in the much simpler case of (pseudo-) Riemannianmetrics, see Chapter 12 of Olver [Ol1995]. It would be interesting to write a program,finer and more efficient than [N2003], which would systematically recognize such rotationformulas in any application of É. Cartan’s equivalence method.l 1 =19