Could an Electromagnetic field produce the "3-body" problem?

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SUMMARY

The discussion centers on the application of electromagnetic (EM) fields in solving the classical three-body problem. By knowing the positions and velocities of bodies along with the EM-field parameters, such as the electric field vector ##\vec{E}(t)## and magnetic field vector ##\vec{B}(t)##, one can compute the dynamics of the system over time. When approximating the speed of light to infinity (##c \to \infty##), the problem simplifies to a classical three-body scenario with an added Coulomb force. However, incorporating magnetic forces based on body positions may compromise the model's invariance under Galileo and Lorentz transformations.

PREREQUISITES
  • Understanding of classical mechanics, specifically the three-body problem
  • Familiarity with electromagnetic theory, including electric and magnetic fields
  • Knowledge of mathematical concepts such as vector calculus and differential equations
  • Basic grasp of invariance principles in physics, including Galileo and Lorentz invariance
NEXT STEPS
  • Explore the mathematical foundations of the three-body problem in classical mechanics
  • Study electromagnetic field theory, focusing on the behavior of ##\vec{E}(t)## and ##\vec{B}(t)##
  • Investigate the implications of non-invariance in physical models, particularly in relation to magnetic forces
  • Learn about numerical methods for simulating dynamical systems influenced by electromagnetic fields
USEFUL FOR

Physicists, mathematicians, and engineers interested in advanced dynamics, particularly those exploring the intersection of classical mechanics and electromagnetic theory.

JorgeM
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If you know positions and velocitys of bodies and EM-field ((##\vec{E}(t)## and ##\vec{B}(t))##) or (EM-poteintial ##\vec{A}(t)## and it's time deriative ##\frac{\partial \vec{A}}{\partial t}##)) at given time, then you can calculate: EM-field, positions and velocities of bodies on any time.

If you make approximation ##c\to \infty##, then it is just like classical 3-bodie problem, but with additional coulomb force.

If you tried to take into account magnetic force as function of positions of bodies, then the model mignt not be galileo invariant nor lorentz invariant.
 
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