Critical Point Classification: Inconclusive Hessian

  • Thread starter Thread starter Gekko
  • Start date Start date
  • Tags Tags
    Hessian
Gekko
Messages
69
Reaction score
0
What is the general approach to take when the Hessian is inconclusive when classifying critical points? ie the determinant = 0?
 
Physics news on Phys.org
If det(H) = 0 and H has both positive and negative eigenvalues at x, then x is a saddle point for the function.

If not...then classifying degenerate critical points [det(H) = 0] becomes quite difficult from what I know. Thom's Splitting Lemma might work. It's sort of a parametrized version of the Morse lemma.
http://en.wikipedia.org/wiki/Splitting_lemma_(functions )

In general, I think it's safe to say that degenerate critical points are annoying haha.
 
Last edited by a moderator:

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
View full post »

Similar threads

Stationary points classification using definiteness of the Lagrangian
Replies
4
Views
2K
Finding the Nature of Critical Points Using Hessian
Replies
8
Views
4K
Why does the Hessian determinant Δ_p = -1 imply that P(0, 0) is a saddle point?
Replies
1
Views
2K
Difference between a hessian and a bordered hessian
Replies
1
Views
13K
Find the extreme values of the polynomial function
Replies
5
Views
2K
Critical points of 4 variables?
Replies
2
Views
3K
Bivariate Function Optimization
Replies
4
Views
2K
Hessian matrix in taylor expansion help
Replies
7
Views
3K
Q about 2nd derivative test for multivariable functions
Replies
8
Views
4K
Is Sin(4πx)Cos(6πx) a Morse Function on a Flat Torus?
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top