In this talk, I will use these concepts to elucidate the main features of resonant dynamics and answer the following questions: Why do different resonances behave similarly? When can resonant trapping occur? Do equilibrium configurations exist and, if so, under what conditions are they stable? I am grateful to both NSF and NASA for supporting this research. ![]() Most of these similarities are enforced by very simple physics, primarily orbital symmetries and the Jacobi Constant. Thus, the mathematical apparatus developed for satellite resonances may be applied to their more exotic non-gravitational counterparts and, conversely, the study of non-gravitational resonances can enhance our understanding of gravitational phenomena. Interestingly, the physics of resonance trapping is largely independent of the details of the resonant force, whether it is the gravity of an orbiting satellite, the gravity of a non-axisymmetric planet, or the Lorentz force from a spinning magnetic field (Hamilton 1994). All of these forces act to drive objects from their initial non-resonant positions into resonant configurations where they can become permanently trapped. For the solution on the tail of the tadpole, we can scale v(x) 1 p(z) z y where is a decaying solution of the second-order equation 00(z) + (p+ 1)jj2p 0 z>0: Let 0(z) sech 1 p(pz) be the unique symmetric solitary wave. The Solar System did not form in its current resonant-rich state though, rather it evolved slowly into this state under the influence of drag forces including planetary tides, planetary migration, Poynting-Robertson drag, plasma drag, and nebular drag. Standing waves bifurcating from the zero resonance Let 2 and consider small values of. ![]()
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