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In this video, I set up and solve the Geodesic Problem on a Sphere. Hamilton's equations of motion, phase space. Hamilton's principle, Lagrange's equations of motion. 5. 68K views 4 years ago Calculus of Variations. Calculus of Variations (8 weeks): Classical problems in the calculus of variations. These results hint onto a possibility for a more general structure on the fibred manifold, than usually assumed. Variational calculus provides a powerful approach for determining the equations of motion constrained to follow a geodesic. The interesting result is that the metric, which induces the connection on the target manifold can change between fibres and these changes are related to the connection on the source manifold. Minimization of functionals, Euler Lagrange equations, sufficient conditions for a minimum, geodesic, isoperimetric and time of transit problems, variational. However, we find that the connection on the source manifold need not be metric.

For the connection on the target manifold, we get the expected result that it is a metric connection. Using the well-established formalism of calculus of variations on fibred manifolds, we solve the weak inverse problem for the equation of geodesic mappings and get a variational equation, which is a consequence of the geodesic mappings equation. Applications 4.1 Great circle distance between any two cities on the Earth References: Livio, M, 2005. I begin by setting up the problem and using the Euler-Lagrange. 1.1 The problem by variational calculus 1.2 The Euler-Lagrange equation The geodesic problem: general formulation Examples 3.1 Plane 3.3 Right circular cylinder 3.5 Hyperbolic paraboloid 3.2 Sphere 3.4 Right circular cone 4. Harmonic mappings are defined between Riemannian manifolds as critical points of the energy functional, on the other hand, geodesic mappings are defined in a more general setting (manifolds with affine connections). 1.5K Share 68K views 4 years ago Calculus of Variations In this video, I set up and solve the Geodesic Problem on a Sphere. In this paper, we study the relation between geodesic and harmonic mappings.