Morse Lemma

Morse's lemma is a statement describing the behavior of a smooth or analytic real function in a neighborhood of a non-degenerate critical point . One of the simple but most important results of Morse theory ; named for the developer of the theory and the American mathematician Marston Morse, who established this result in 1925 .

Content

Wording

Let be $f:\mathbb {R} ^{n}\to \mathbb {R}$ ${\ displaystyle f: \ mathbb {R} ^ {n} \ to \ mathbb {R}}$ - class function $C^{r+2}$ ${\ displaystyle C ^ {r + 2}}$ where $r\geq 1$ ${\ displaystyle r \ geq 1}$ having a point $0\in \mathbb {R} ^{n}$ ${\ displaystyle 0 \ in \ mathbb {R} ^ {n}}$ its non-degenerate critical point, that is, at this point the differential ${\frac {\partial f}{\partial x}}$ ${\ displaystyle {\ frac {\ partial f} {\ partial x}}}$ vanishes, and the Hessian ${\Bigl |}{\frac {\partial ^{2}f}{\partial x^{2}}}{\Bigr |}$ ${\ displaystyle {\ Bigl |} {\ frac {\ partial ^ {2} f} {\ partial x ^ {2}}} {\ Bigr |}}$ different from zero. Then in some neighborhood $U$ ${\ displaystyle U}$ points $0$ ${\ displaystyle 0}$ there is such a system $C^{r}$ ${\ displaystyle C ^ {r}}$ -smooth local coordinates (map) $(x_{1},x_{2},\ldots ,x_{n})$ ${\ displaystyle (x_ {1}, x_ {2}, \ ldots, x_ {n})}$ starting at point $0$ ${\ displaystyle 0}$ that for everyone $x\in U$ ${\ displaystyle x \ in U}$ equality holds ^[1]

f(x)=f(0)-x_{1}^{2}-\dots -x_{k}^{2}+x_{k+1}^{2}+\dots +x_{n}^{2}

{\ displaystyle f (x) = f (0) -x_ {1} ^ {2} - \ dots -x_ {k} ^ {2} + x_ {k + 1} ^ {2} + \ dots + x_ { n} ^ {2}}

.

Moreover, the number $k$ ${\ displaystyle k}$ defined by the signature of the quadratic part of the germ $f$ ${\ displaystyle f}$ at the point $0$ ${\ displaystyle 0}$ is called the critical point index $0$ ${\ displaystyle 0}$ this function is a special case of the general concept of the Morse index .

Variations and generalizations

Tujron's Theorem

In the vicinity of the critical point $0$ ${\ displaystyle 0}$ finite multiplicity $\mu$ ${\ displaystyle \ mu}$ there is a coordinate system in which a smooth function $f(x)$ ${\ displaystyle f (x)}$ has the form of a polynomial $P_{\mu +1}(x)$ ${\ displaystyle P _ {\ mu +1} (x)}$ degrees of $\mu +1$ ${\ displaystyle \ mu +1}$ (as $P_{\mu +1}(x)$ ${\ displaystyle P _ {\ mu +1} (x)}$ can take the taylor polynomial function $f(x)$ ${\ displaystyle f (x)}$ at the point $0$ ${\ displaystyle 0}$ in original coordinates). In the case of a non-degenerate critical point, the multiplicity $\mu =1$ ${\ displaystyle \ mu = 1}$ , and the Toujron theorem turns into Morse's lemma ^[1] ^[2] .

Morse lemma with parameters

Let be $f(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{m}):\mathbb {R} ^{n+m}\to \mathbb {R}$ ${\ displaystyle f (x_ {1}, \ ldots, x_ {n}, y_ {1}, \ ldots, y_ {m}): \ mathbb {R} ^ {n + m} \ to \ mathbb {R} }$ - smooth function with origin $0$ ${\ displaystyle 0}$ its critical point, non-degenerate in variables $x_{1},\ldots ,x_{n}$ ${\ displaystyle x_ {1}, \ ldots, x_ {n}}$ . Then in a neighborhood of the point $0$ ${\ displaystyle 0}$ there are smooth coordinates in which

f(x,y)=\alpha _{1}x_{1}^{2}+\cdots +\alpha _{n}x_{n}^{2}\,+\,f_{0}(y_{1},\ldots ,y_{m}),\quad \alpha _{i}=\pm 1,

{\ displaystyle f (x, y) = \ alpha _ {1} x_ {1} ^ {2} + \ cdots + \ alpha _ {n} x_ {n} ^ {2} \, + \, f_ {0 } (y_ {1}, \ ldots, y_ {m}), \ quad \ alpha _ {i} = \ pm 1,}

Where $f_{0}$ ${\ displaystyle f_ {0}}$ Is some smooth function. This statement allows us to reduce the study of the singularity (critical point) of the function of $n+m$ ${\ displaystyle n + m}$ variables to the study of the features of a function of a smaller number of variables (namely, of the number of variables equal to the corank of the Hessian of the original function) ^[1] .

The proof of this statement can be carried out by induction on n using the Hadamard lemma or in another way ^[1] .

Notes

↑ ¹ ² ³ ⁴ Arnold V.I., Varchenko A.N., Huseyn-Zade S.M. Features of differentiable mappings.
↑ Samoilenko A. M. On the equivalence of a smooth function to a Taylor polynomial in a neighborhood of a critical point of finite type, - Funkts. Analysis and its adj., 2: 4 (1968), pp. 63-69.

Literature

Arnold V.I., Varchenko A.N., Huseyn-Zade S.M. Features of differentiable mappings.
Zorich V.A. Mathematical analysis.
Hirsch M. Differential topology.
Takens F. A note on sufficiency of jets. - Inventiones Mathematicae, vol. 13, no 3, 1971, pp. 225-231.
Samoilenko A. M. On the equivalence of a smooth function to a Taylor polynomial in a neighborhood of a critical point of finite type, - Funkts. Analysis and its adj., 2: 4 (1968), pp. 63-69.
Darinsky B. M., Sapronov Yu. I., Tsarev S. L. Bifurcation of extremals of Fredholm functionals, - CMFD, 12, M., 2004, p. 3-140.

[autogenerated1-1] ¹ ² ³ ⁴ Arnold V.I., Varchenko A.N., Huseyn-Zade S.M. Features of differentiable mappings.

[2] Samoilenko A. M. On the equivalence of a smooth function to a Taylor polynomial in a neighborhood of a critical point of finite type, - Funkts. Analysis and its adj., 2: 4 (1968), pp. 63-69.