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On the lowest energy state of a collisionless self-gravitating system under phase space volume constraints. (English) Zbl 0682.70009

Summary: The problem of minimizing the energy of a collisionless self-gravitating system under the only constraint of phase space volume conservation is reconsidered. We bring about essentially three new results: (i) we establish a lower bound on the energy by using arguments which are both simpler and more general than those of other authors; (ii) we prove that the lowest energy state is necessarily spherically symmetric (in the physical space), and (iii) we show that, if we fix also the total angular momentum J of the system, then there is no lowest energy state satisfying this further constraint if \(J\neq 0\).

MSC:

70F10 \(n\)-body problems
70F15 Celestial mechanics
70G10 Generalized coordinates; event, impulse-energy, configuration, state, or phase space for problems in mechanics
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