I realized that the geodesic on a linear surface is just a straight line so this is just a matter of minimizing the distance between $A$ and the line of intersection plus the distance between $B$ and the line of intersection. try finding the geodesic, or the shortest curve, i.e. Minimization of functionals, Euler Lagrange equations, sufficient conditions for a minimum, geodesic, isoperimetric and time of transit problems, variational. I have a feeling that there isn't any nice solution but I'm hoping that there are some simplifications. Calculus of Variations is used as a device for optimization problems that in. I would love help setting up the general problem or links to papers or articles that talk about this. ![]() It was an unpleasant incident, but one of great value to mathematics for the problems being argued about led directly to the founding of the calculus of variations. A short history of Calculus of Variations Problems from Geometry Necessary condition: Euler-Lagrange equation Problems from mechanic. I now cite the instructions and answer as found on the book. ![]() My inclination is that it boils down to a boundary value problem in the calculus of variations and finding the geodesic subject to the constraint that the end point is on the line of intersection but I cannot find a way to set the problem in general. The solution of the geodesic problems, with one of the above two methods, includes evaluating elliptic integrals or solving differential equations using: (i) approximate analytical methods, e.g., Vincenty (1975), Holmstrom (1976), Pittman (1986), Mai (2010), Karney (2013) or (ii) numerical methods, e.g., Saito (1979), Rollins (2010), Sjberg (20. Jacob Bernoulli posed isoperimetric problems to Johann Bernoulli and a bitter dispute arose between the two brothers on these problems which Varignon also became involved in. I am interested in the development of this problem which should be done by calculus of variations. If $A$ is a point on $U$ and $B$ is a point on $V$ how do you find the shortest path(s) between the points? ![]() Since the surfaces are distinct and intersect the intersection must be a line. Suppose we have two distinct surfaces $U$ and $V$ in $R^3$ that intersect. 10 CALCULUS OF VARIATIONS A basic problem in semi-Riemannian geometry is to measure the change in arc length of a curve segment under small displacements. I've become interested in finding geodesic on the intersection of riemann manifolds however it turns out much of the literature is way above my head so I'm looking into simpler cases.
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