Morse index theorem - Revision history

Vipul: 2 revisions

2008-05-18T19:50:01Z

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Vipul: /* Setup */

2007-06-23T17:27:03Z

Setup

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	Let <math>P</math> be a <math>C^\infty</math>-map from <math>[0,T]</math> into the collection of self-adjoint linear transformations on <math>\R^n</math>.		Let <math>P</math> be a <math>C^\infty</math>-map from <math>[0,T]</math> into the collection of self-adjoint linear transformations on <math>\R^n</math>.

	Let <math>U</math> be the unique <math>C^\infty</math>-map from <math>[0,T]\to ~~Home~~(\R^n,\R^n)</math> such that <math>U\\ + PU = 0</math>.		Let <math>U</math> be the unique <math>C^\infty</math>-map from <math>[0,T]\to Hom(\R^n,\R^n)</math> such that <math>U'' + PU = 0</math>.

	Define:		Define:

Vipul at 17:26, 23 June 2007

2007-06-23T17:26:13Z

New page

{{index theorem}}

==Statement==

===Setup===

Let <math>\R^n</math> be Euclidean space, and let <math>G</math> be the linear space of piecewise <math>C^\infty</math>-maps from <math>[0,T]</math> to <math>\R^n</math>. Let <math>H</math> denote the subspace of <math>G</math> comprising maps which are zero at the endpoints (viz maps <math>u</math> satisfying <math>u(0) = u(T) = 0</math>).

Let <math>P</math> be a <math>C^\infty</math>-map from <math>[0,T]</math> into the collection of self-adjoint linear transformations on <math>\R^n</math>.

Let <math>U</math> be the unique <math>C^\infty</math>-map from <math>[0,T]\to Home(\R^n,\R^n)</math> such that <math>U\\ + PU = 0</math>.

Define:

* The multiplicity of <math>t \in [0,T]</math> is the nullity of <math>U(t)</math>
* <math>t \in [0,T]</math> is said to be a '''focal point''' if its multiplicity is positive

Define the [[index form]] corresponding to <math>P</math> as the following bilinear form on <math>H</math>:

<math>I(u,v) = \int_0^T [(u',v') - (Pu,v)] dt</math>

===Statement part===

* The index <math>i(I)</math> (viz the [[index of a bilinear form|index]] of the index form) is the sum of multiplicities of all focal points in the open interval <math>(0,T)</math>
* The nullity <math>n(I)</math> (viz the [[nullity of a bilinear form|nullity]] of the index form) is the multiplicity of <math>T</math>.