Index form

Definition

Given data

Let $\mathbb {R} ^{n}$ be Euclidean space, and let $G$ be the linear space of piecewise $C^{\infty }$-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 $C^{\infty }$-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: $I(u,v)=\int _{0}^{T}[(u',v')-(Pu,v)]dt$ where $(,)$ is the inner product on $\mathbb {R} ^{n}$.