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. of the Euler-Lagrange equations, and finding explicit solutions of classical problems, like the Brachistochrone problem, and exploring applications to image. I have a feeling that there isn't any nice solution but I'm hoping that there are some simplifications. I would love help setting up the general problem or links to papers or articles that talk about this. 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. Figure 1: The isoperimetric problem and the geodesic on a sphere. We can say, he started to develop the general theory of calculus of variations. The calculus of variations originated in problems to maximize or minimize certain. Hamilton's equations of motion, phase space. Leonhard Euler got interested in geodesic curves and isoperimetic problems and started to develop the theory of minimal curves and surfaces )1744). Hamilton's principle, Lagrange's equations of motion. 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? Calculus of Variations (8 weeks): Classical problems in the calculus of variations. formal apparatus of the calculus of variation-Lagranges equation, Legendres, Jacobis, Weierstrass conditions and Hilberts independent integral-will be found in these three Parts, packed together in a nutshell as it were. Since the surfaces are distinct and intersect the intersection must be a line. ![]() ![]() To find the geodesics on a surface is a variational problem involving the. Suppose we have two distinct surfaces $U$ and $V$ in $R^3$ that intersect. The calculus of variations is a powerful technique for the solution of. A geodesic is a special curve that represents the shortest. 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. 54K views 5 years ago Calculus of Variations In this short (hehe) video, I set up and solve the Geodesic Problem on a Plane.
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