Morse lemma

When the function F(x) has a critical point at x = 0, the transformation defined above, (2.28), does not determine a one-to-one change of variables nearby the point x = 0, because the first row of the Jacobian matrix vanishes and, consequently, so does its determinant. As a result, such a function cannot be represented as F(x ) = x/ in the vicinity of its critical point. In a case, however, when the critical point x = 0 is structurally stable (a precise criterion will be provided in further part of this section), the function F may be reduced in the vicinity of this point to a simpler form. 

The method of reduction of the form of the function F(x) in the case of a structurally stable critical point will be discussed for a function F dependent on two variables. The presented results are a special case of the Morse lemma. 

Let the function F(x), x = (x, y), having at x = (0,0) a critical point, have the following general form  

Indeed, at x = 0 a function F of the form (2.30) has a critical point, since at this point vanish first-order partial derivatives  

We shall assume about the function V that in the neighbourhood of the point x = 0 the Hessian matrix Vu has a non-zero determinant 

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An important stage in the construction of the ordinary Morse theory is the well-known Morse lemma. It asserts that in some open neighbourhood of a nondegenerate critical point Xq of a Morse function /, there always exist local regular coordinates yi,..., ym such that the function / is written in the form... 

Proof The analogue to this Lemma in the ordinary Morse theory is well known, but in our case the proof is more delicate, since here we deal with the integral f (and not merely with a smooth function), and therefore we should essentially use the conditons of Theorem 2.1.3. An arbitrary smooth perturbation of an integral... 

This is due to the celebrated Morse s bifurcation lemma (Morse, 1931) around the so-called critical points of the original function, rj c,x), where it actually equivalents the original function with the family of function (Putz et ah, 2011b)... 

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