Index form: Difference between revisions

Latest revision as of 19:47, 18 May 2008

Definition

Given data

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

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

Definition part

Then the index form associated with $P$ is the bilinear form on $H$ defined as follows: $I (u, v) = \int_{0}^{T} [(u^{'}, v^{'}) - (P u, v)] d t$ where $(,)$ is the inner product on $R^{n}$ .

@@ Line 5: / Line 5: @@
 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>).
-Then the '''index form''' is the bilinear form on <math>H</math> defined as follows: <math>I(u,v) = \int_0^T [(u',v') - (Pu,v)] dt</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>.
+===Definition part===
+Then the '''index form''' associated with <math>P</math> is the bilinear form on <math>H</math> defined as follows: <math>I(u,v) = \int_0^T [(u',v') - (Pu,v)] dt</math> where <math>(,)</math> is the inner product on <math>\R^n</math>.