The Algorithm

The purpose of this code is to provide a consistent numerical method to find the minimum of the functional

\[L[u] = \int_0^1\sqrt{2(E-V(u(\tau)))}\|u'(\tau)\| d\tau\]

over the class of Lipschitz curves joining \(q_a\) to \(q_b\) in \(\mathbb R^d\). Here we will assume that \(V \in C^2(\mathbb R^d)\) and \(E > \| V \|_{\infty}\). The Maupertuis principle states that by computing this minimum, we consequently compute solutions to

\[q''(t) = -\nabla V(q(t))\]

subject to \(q(0) = \xi_1\) and \(q(T) = \xi_2\) for some \(T\) to be determined as a function of \(E\). The MODOI software package uses effective medium theory to compute the value of V based on the atomistic description of the system.

For full details consult Chapter 4 of Microscopic Hamiltonian Systems and their Effective Description pdf.