Geodesic
On the theory of degenerate quadratic forms in the classical calculus of variations
Izvestiya. Mathematics , Tome 45 (1995) no. 3, pp. 433-476.

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This paper investigates the property of having definite sign for a quadratic form in the calculus of variations that does not satisfy the strong Legendre condition. New necessary conditions (the condition $\mathscr A$) of pointwise nature are obtained for the index of a form to be finite. Under the assumption of the strong condition $\mathscr A$ a formula is obtained for the index of a form. 
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A. V. Arutyunov. On the theory of degenerate quadratic forms in the classical calculus of variations. Izvestiya. Mathematics , Tome 45 (1995) no. 3, pp. 433-476. http://geodesic.mathdoc.fr/item/IM2_1995_45_3_a0/

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