Jacobi equation

The Jacobi equation is the linearization of an Euler-Lagrange equation about a stationary path. Its solutions describe infinitesimal one-parameter families of stationary paths.

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- Jacobi identity for the second variation Jacobi equation

Jacobi identity for the second variation ( $P φ^{'2} + Q φ^{2} = P (φ^{'} - w^{'} φ / w)^{2} + (P w^{'} φ^{2} / w)^{'}$ )

If

w

is a nonvanishing solution of

- (P w^{'})^{'} + Qw = 0

, expanding the right-hand side gives the displayed identity. Integrating expresses a scalar second variation as a square plus a boundary term. A simple zero of

w

at a fixed endpoint causes no difficulty for Lipschitz endpoint-vanishing variations, since

φ = O (x - a)

makes the boundary term tend to zero. This identity is the scalar form of a ground-state transform for a weighted Dirichlet energy.

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