By Stuart S. Antman (auth.), Gerald Kaiser, Jerrold E. Marsden (eds.)
Read or Download Geometric Methods in Mathematical Physics: Proceedings of an NSF-CBMS Conference Held at the University of Lowell, Massachusetts, March 19–23, 1979 PDF
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Additional info for Geometric Methods in Mathematical Physics: Proceedings of an NSF-CBMS Conference Held at the University of Lowell, Massachusetts, March 19–23, 1979
Example text
In the level sets of the H a m i l t o n i a n g e n e r a t i n g function for the symmetry group). In general such singu- larities need not be "conical" but could instead be of higher order. The conical singularities however seem to be common in p r a c t i c e equations arising in physics. for 48 As a simple example m o v i n g in ~ 3 from mechanics we consider N point particles The phase space for this system is T * ~ N ~ ~ 6 N and the moment map for (say) the usual S0(3) action on this space is simply the total angular m o m e n t u m vector ~.
The case of perfect fluids coupled to gravity has heretofore seemed quite different from that of pure gravity or of gauge fields coupled to gravity. D'Eath 12 showed that one could always solve the constraint equations algebraicly for certain of the fluid variables. Thus one does not encounter linearization instabilities even for highly symmetrical spacetimes such as the Robertson-Walker models considered in detail by D'Eath. A similar conclusion is implicit in the Hamiltonian treatment of perfect fluid dynamics given by the author 13.
The out in [I0]. certain H(a) value problem. 6) with of the vector the buckling Thus = 0 to vanish. is a critical to be zero. from the replacement that of its zeros only forces the vanishing point result we must it can change We assert = 0, n'(a) ~, at this critical The treatment that branch. 3) and the left, n ~ trivial sets of solution branch preserve variable with the property this function compressed not much worse The problem of simplifications. the deformation For becomes very large (in a way that can be made precise) every buckled state has at least two axes of symmetry and preserves the nodal properties of ~,~ and the curvature 6.
