Index form

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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') - (Pu,v)] dt where (,) is the inner product on \R^n.