# First variation formula

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## Statement

### Notation

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

$\frac{1}{2} \frac{dE(\overline{\alpha}(u))}{du} = - \sum_{i=1}^{k-1} - \int_0^1 dt$