Index form

Definition

Given data

Let ${\mathbb{R}}^n$ be Euclidean space, and let $G$ be the linear space of piecewise $C^{\mathrm{\infty }}$-maps from $[0,T]$ to ${\mathbb{R}}^n$. 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 $C^{\mathrm{\infty }}$-map from $[0,T]$ into the collection of self-adjoint linear transformations on ${\mathbb{R}}^n$.

Definition part

Then the index form associated with $P$ is the bilinear form on $H$ defined as follows: $I(u,v)=\underset{0}{\overset{T}{\int }}[(u^{\prime },v^{\prime })-(Pu,v)]dt$ where $(,)$ is the inner product on ${\mathbb{R}}^n$.