6 Calculus of variation

Definition 6.1 (Calculus of variation) Suppose \(M\) is a manifold, and \(L: TM \to \mathbb R\) is a smooth functional. A curve \(\gamma: \mathbb R \to M\) is said to be a critical point of calculus of variations if in local coordinates \((U, q, \dot q)\), \[ \frac{d}{dt} \left ( \frac{\partial L}{\partial \dot q} \circ \gamma_* \right) - \frac{\partial L}{\partial q} \circ \gamma_* = 0 \] where \[ \begin{aligned} \frac{\partial L}{\partial q} &= \left ( \frac{\partial L}{\partial q^1}, \quad \dots, \quad \frac{\partial L}{\partial q^n} \right ) \\ \gamma_* &= (\gamma, \gamma') \end{aligned} \]