Second variation

Remarks

Some books put a factor $\frac{1}{2}$ in front, so that $J(y_0 + \varepsilon h) = J(y_0) + \varepsilon \, \delta J(y_0)[h] + \varepsilon^2 \, \delta^2 J(y_0)[h] + o(\varepsilon^2)$, where $\delta J$ is the first variation.

Example: let $J(y) = \int_0^\pi (y'^2 - y^2) \, dx$ and $y_0 = 0$. Then $J(\varepsilon h) = \varepsilon^2 \int_0^\pi (h'^2 - h^2) \, dx$, so $\delta^2 J(0)[h] = 2 \int_0^\pi (h'^2 - h^2) \, dx$. This agrees with the formula, since $L_{pp} = 2$, $L_{yp} = 0$ and $L_{yy} = -2$.

Non-example: let $J(y) = \int_0^1 \abs{y'} \, dx$ on $C^1([0, 1])$ and $y_0 = 0$. Then $J(\varepsilon h) = \abs{\varepsilon} \int_0^1 \abs{h'} \, dx$, which is not differentiable at $\varepsilon = 0$ when $h$ is not constant. So the second variation at $0$ does not exist in such a direction $h$; the integrand $\abs{p}$ is not of class $C^2$.

If $h(a) = h(b) = 0$ and $x \mapsto L_{yp}(x, y_0(x), y_0'(x))$ is $C^1$, an integration by parts gives $\delta^2 J(y_0)[h] = \int_a^b \big( P h'^2 + Q h^2 \big) \, dx$ with $P = L_{pp}$ and $Q = L_{yy} - \frac{d}{dx} L_{yp}$.