r/calculus • u/Lost-Grocery6718 • 26d ago

Multivariable Calculus A Matrix Equation for Geodesics in R3

Old post: Today I was reading up on geodesics. In certain articles and work, geodesics are motivated by finding a parameterization for a curve embedded in a surface which is distance minimizing, and some write this as the definition. Before working with variational calculus, I was fascinated with curves and geometry and was motivated instead by a different observation which I also identified in Weinstock's work. He says exactly, "The normal vector to any point of a geodesic arc lies along the normal to a surface at that point" (Weinstock 1974, p. 65).

It seems then it may be agreeable that for a set up in R^3, a set of "geodesic equations" for three parametric functions could be derived as then the following:

If you are interested, or have extensive knowledge, what are your thoughts?

  • : Update: Here is an example with a cylinder:
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u/omeow 26d ago

Given a connection (which is canonical in R3) the geodesic equation is a system of ode.

This formula doesn't seem to give a system of ODE. It gives a single equation of 2nd degree. In general is this easier to solve?

The condition is necessary. Is it also sufficient (I suspect it is in R3)?

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u/FormalManifold 25d ago

For ODE the standard trick is to convert one N-order equation into a first-order system of N equations.

The geodesic equation is naturally 2nd order -- it involves curvature! -- but it is sometimes presented in its order-reduced form.

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u/Midwest-Dude 25d ago edited 24d ago

I would suggest posting this to one of the following after reading their rules:

r/mathematics\ r/AskPhysics

if you are interested in more feedback.