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I'm searching for the minimum path between the minima of a potential surface that is already known on a grid.

example

(source: http://www.math.nus.edu.sg/~matrw/string/)

Any point on the path is at an potential minimum in all directions perpendicular to the path.

Is there any SciPy method or any other python package to compute this path? I'm not looking for a method that can search on an unknown surface.

edit: I'm searching the path with the lowest potential barrier.

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  • $\begingroup$ As soon, as I reach the reputation, I can provide an image to clarify. $\endgroup$
    – tmartin
    Jul 30, 2013 at 14:35
  • $\begingroup$ When you say the data is already on a grid, does that mean you're trying to find this path on the gridded data with minimal further computation? $\endgroup$
    – Richard
    Jan 9, 2019 at 4:22

2 Answers 2

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I don't know if you are aware of it but you can find some MATLAB examples (not Python, though) for the Mueller potential at Eric Vanden-Eijnden's string method page.

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What you are looking for is a way to find a geodesic on a known surface with a known metric. This is a classical geometry problem. The place to look for algorithms is in books on computational geometry.

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  • $\begingroup$ As far as I understand the term geodesic, this is the shortest path on the surface connecting the two points. Considering a surface with a shallow, but curvy basin, the geodesic would shortcut the line which describes the condition I postulated in the question. $\endgroup$
    – tmartin
    Jul 31, 2013 at 15:22
  • $\begingroup$ Oh, I see, I misunderstood the question. Your path is always parallel to the gradient of the function whose graph is the surface you are considering. But then there may be many such paths between any two points -- which one do you select? $\endgroup$
    – Wolfgang Bangerth
    Jul 31, 2013 at 19:58
  • $\begingroup$ In typical problems, only the path with extremal action $S$ ($\delta S = \delta \int L(q, \dot q, t)\,\mathrm{d}t$ = 0$) is realised by nature. With zero variation at initial&final times, these are the paths obtained from the Euler-Lagrange equations. $\endgroup$
    – AlexE
    Aug 2, 2013 at 20:17

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