Geometric conditions for the positive definiteness of the second variation in one-dimensional problems

The superscripts D and N used to denote the space of admissible perturbations ${{\mathcal{C}}^{\text{D}}}$ and ${{\mathcal{C}}^{\text{N}}}$ respectively refer to the Dirichlet and Neumann boundary conditions that naturally arise when minimising $\mathcal{E}$ with respectively fixed and free boundaries.

Finding stationary functions for the functional $\mathcal{E}$ amounts to solving the boundary value problem (15). We had noted that in the case of fixed boundaries, this BVP has Dirichlet boundary conditions while for free boundaries, the BVP has Neumann boundary conditions. Something similar happens here. For fixed boundaries, the stability can be assessed by solving a Sturm–Liouville problem with Dirichlet boundary conditions while for free boundaries, the S–L problem has Neumann boundary conditions.

This is easily proven by considering  $\begin{array}{*{35}{l}} \mathcal{S}{{\theta}^{\prime}} & = & -{{\theta}^{\prime \prime \prime}}-\frac{{{\text{d}}^{2}}V}{\text{d}{{\theta}^{2}}}{{|}_{\theta (s)}}\,{{\theta}^{\prime}}\overset{{}}{{\overset{(16)}{{=}}\,}}\,{{\left(\frac{\text{d}V}{\text{d}\theta}{{|}_{\theta (s)}}\right)}^{\prime}}-\frac{{{\text{d}}^{2}}V}{\text{d}{{\theta}^{2}}}{{|}_{\theta (s)}}\,{{\theta}^{\prime}}=0. \end{array}$

If P  >  0, then η starts at $\left({{T}_{0}},\,{{P}_{0}}\right)$ and end at $(T,\,P)$. If P  <  0, then η starts at $(T,\,P)$ and end at $\left({{T}_{0}},\,{{P}_{0}}\right)$. Either way the length formulae below hold.

In this case, we can substitute variable in the integral appearing in the RHS of (45) according to $\text{d}t=\frac{\text{d}\theta}{\text{Sign}\left[{{\theta}^{\prime}}(s)\right]\sqrt{2}\,\sqrt{E-V(\theta )}}$.

These two functions can not be linearly dependent since ${{\theta}^{\prime}}(d)\ne 0$ while $\mu (d;d)=0$.

That is assuming that U  >  0, if U  <  0 the solution starts at $\arccos [v-e]$.