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| Author(s): |
Daniel Câmara de Souza
Total Authors: 1
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| Document type: | Doctoral Thesis |
| Press: | São Paulo. |
| Institution: | Universidade de São Paulo (USP). Instituto de Física (IF/SBI) |
| Defense date: | 2014-12-18 |
| Examining board members: |
Jayme Vicente de Luca Filho;
Domingos Humberto Urbano Marchetti;
Clodoaldo Grotta Ragazzo;
Savio Brochini Rodrigues;
Ricardo Luiz Viana
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| Advisor: | Jayme Vicente de Luca Filho; Coraci Pereira Malta |
| Abstract | |
We study the Wheeler-Feynman electrodynamics using a variational principle for an action functional coupled to a finite boundary value problem. For piecewise C2 trajectories, the critical point condition for this functional gives the Wheeler-Feynman equations of motion in addition to a continuity condition of partial moments and partial energies, known as the Weierstrass-Erdmann corner conditions. In the simplest case, for the boundary value problem of shortest length, we show that the critical point condition reduces to a two-point boundary value problem for a state-dependent mixed-type neutral differential-delay equation. We solve this special problem numerically using a shooting method and a fourth order Runge-Kutta. For the cases where the boundary segment has discontinuous velocities we developed a technique to solve the Weierstrass-Erdmann corner conditions and the two-point boundary value problem together. The trajectories with discontinuous velocities presupposed by the variational method were verified by numerical experiments. In a second development, for the harder case with boundaries of arbitrary length, we implemented a method of minimization with weak gradient for the variational principle quoted above. Two numerical methods were implemented in MATLAB to find solutions of the two-body electromagnetic problem. The first combines the finite element method with Newtons method to find the solutions that vanish the weak gradient. The second uses the method of steepest descent to find the solutions that minimize the action. In both methods the trajectories are approximated within a finite-dimensional space generated by a Galerkian that supports discontinuous velocities. Many tests and numerical experiments were performed to verify the convergence of the numerically calculated trajectories; also were compared the values of the functional computed numerically with some known analytical results on circular orbits. (AU) | |
| FAPESP's process: | 10/16964-0 - Variational Electrodynamics and Electromagnetic Problem of Two and Three Bodies |
| Grantee: | Daniel Câmara de Souza |
| Support Opportunities: | Scholarships in Brazil - Doctorate |