Timeline for Why trace is more natural than (preferred to) determinant for smooth map $f:M\to N$?
Current License: CC BY-SA 4.0
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| Dec 3, 2020 at 21:16 | history | edited | LSpice | CC BY-SA 4.0 |
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| Dec 3, 2020 at 21:14 | comment | added | LSpice | See @D.Savitt's wonderful answer to the analogous question in representation theory. | |
| Dec 3, 2020 at 21:03 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Nov 3, 2020 at 20:50 | comment | added | Willie Wong | In the case of elastodynamics, see numdam.org/item/?id=AIHPA_1998__69_3_275_0 There are probably also references you can find for the static case. | |
| Nov 3, 2020 at 18:51 | answer | added | Liviu Nicolaescu | timeline score: 1 | |
| Nov 3, 2020 at 17:48 | comment | added | C.F.G | @WillieWong: Thanks for reference. Is the special case of $L(I_1, I_2, I_3)$ known in elasticity? Like the kinetic energy in physics? | |
| Nov 3, 2020 at 17:36 | comment | added | Sebastian | @C.F.G.: the integrand is independent of the metric $g$ on $M$, and using $g=f^*h$ the integral is the area (or volume) of $f(M)\subset N$. The critical points of this functional are minimal surfaces, which are well-studied objects in differential geometrie for centuries. | |
| Nov 3, 2020 at 17:11 | comment | added | Willie Wong | In the case where $(M,g)$ is Lorentzian, I treated some general properties of the Euler Lagrange equation in iopscience.iop.org/article/10.1088/0264-9381/28/21/215008 ; the case that Sebastian mentioned is a special case of the Born-Infeld model, and is related to minimal surfaces. | |
| Nov 3, 2020 at 17:02 | comment | added | Willie Wong | In the elliptic case, general functionals of the form $\int L(I_1, I_2, I_3)$ is considered in elasticity. Here the dimensions of $M, N$ are both 3, and $I_1 = \mathrm{tr} f^*h$, $I_3 = \det f^*h$, and $I_2$ is the quadratic invariant. | |
| Nov 3, 2020 at 15:49 | comment | added | C.F.G | @Sebastian: Any reference? | |
| Nov 3, 2020 at 12:45 | comment | added | Sebastian | I am sure some experts consider $A(f)=\int_M\sqrt{\det_g(f^*h)}dvol_g.$ | |
| Nov 3, 2020 at 12:44 | comment | added | Asvin | A related question I have had for a while that might help (or not): In algebraic geometry and number theory, it is very easy to use the trace and determinant of a linear map (often as the trace and norm of an element in some ring) and I always wondered why other symmetric polynomials in the eigenvalues didn't get considered. The answer is usually that we can somehow exploit the linearity or multiplicativity in some fashion and the other symmetric polynomials in the eigenvalues don't have such nice functionality. In your case, maybe being additive is a really nice property for the trace ? | |
| Nov 3, 2020 at 10:56 | history | asked | C.F.G | CC BY-SA 4.0 |