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

77+ 1 2∑kH k l 1 ,y l 2 Hl 2k15− 1 2∑kH k l 2 ,y l 1 Hl 2k16+ 1 6 Hl 2l 2 ,y l 2 Hl 2l1c−− 1 6 Hl 1l 1 ,y l 1 Hl 2l2− 1e 3 Ll 2l2 ,l 2 ,x Hl 2l1+ 1d 3 Ll 1l1 ,l 1 ,x Hl 2l2f−∑− 1 3 Gl 2H l 2l1M l2 ,l 2+ 1n 3 Gl 1H l 2l2M l1 ,l 1− 2t 3+ 2 3+ 1 6+ 1 6∑p∑p∑pG p H l 2l2M l1 ,pu− 1 2H l 2l1H l 2 p L p l 2 ,l 2− 1 6qH l 2l2H p l 1L l 1l1 ,px+∑ ∑k∑pppH l 2kH p l 1L k l 2 ,pG p H l 2l1M l2 ,ppH l 2l2H l 1 p L p l 1 ,l 1− 1 6w+ 1 2∑po∑ ∑kp+H l 2l1H p l 2L l 2l2 ,pH l 2kH p l 2L k l 1 ,pr+v++ 2 ∑ kL l 2k,l 2 ,y l 1 Gk 17− 2 ∑ kL l 2k,l 1 ,y l 2 Gk 18− 2 ∑ kM k,l1 ,x G k 19++ ∑ k− ∑ p− 2 ∑ kG k H l 2l1M l1 ,l 2− ∑okG l 2H p l 2M l1 ,p∑ps− ∑ kG k L p k,l 2L l 2l1 ,pkG k H l 2l2M k,l1 + ∑ G l 2H p l 1M l2 ,p −pum∑G k H p k M l 1 ,p + 2 ∑ ∑G k L p k,l 1L l 2l2 ,ppk p.In conclusion, there is exact coincidence with the subgoal (4.29). Theproof that the first family (3.112) 1 of compatibility conditions of the secondauxiliary system (3.99), (3.100), (3.101) and (3.102) are a consequence of(I), (II), (III) and (IV) of Theorem 1.7 (3) is complete. Granted that thetreatment of the other three families of compatibility conditions (3.112) 2 ,(3.112) 3 and (3.112) 4 is similar (and as well painful), we consider that theproof of the equivalence between (1) and (3) in Theorem 1.7 is complete,now.20g−§5. GENERAL FORM OF THE POINT TRANSFORMATIONOF THE FREE PARTICLE SYSTEMThis section is devoted to the exposition of a complete proof ofLemma 3.32. To start with, we must develope the fundamental equations(3.10), for j = 1, . . .,m. Recalling that the total differentiation