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Diffgeom β

First variation formula



Let M be a Riemannian manifold and p,q \in M. Let \Omega = \Omega(M;p,q) be the path space of M, viz the space of piecewise smooth paths from p to q.

Let the points of non-smoothness by 0=t_0 < t_1 < t_2 \ldots < t_k. Let \alpha be a variation of \omega, and W be its variation vector field. Let V_t = d\omega/dt wherever \omega is differentiable. For each t_i where \omega is not smooth, let \Delta_{t_i}V = V_{t_i^+} - V_{t_i^-}.

Denote by \overline{\alpha} the map sending u to the curve t \mapsto \alpha(u,t).

The formula

\frac{1}{2} \frac{dE(\overline{\alpha}(u))}{du} = - \sum_{i=1}^{k-1} <W_t, \Delta_t V> - \int_0^1 <W_t,A_t> dt