By M. Francaviglia

Supplying a logically balanced and authoritative account of the various branches and difficulties of mathematical physics that Lagrange studied and built, this quantity provides up to date advancements in differential goemetry, dynamical structures, the calculus of diversifications, and celestial and analytical mechanics.

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**Example text**

Vishik, Attractors their dimension, [2] P. Berge, Turbulence spatiaux, [3] of partial differential evolution equations and Russian Math. Survey 38, 1983, 151-213. en convection de Rayleigh-Benard : du chaos tempore! aux effets Math. Model, and Numer. Anal. 2 3 , 1989, 371-378. EL Brezis and T. , T M A , 4 , 1980, 677-681. [4] P. Constantin, Collective derivatives, [5] L°° estimates for families P. Constantin and C. Foias, Global Lyapunov dimension of functions with orthonormal Indiana Univ. Math.

15). 14) by setting β = 0. 14) a and ν by a' = (a + β ) , 2 2 1/2 v' = ^(a + 2 βψ . 18) It is shown in Sec. 18) (ao depends on v, see Sec. 6). e. 6). g. 1). e. is proportional to the area of a sector of an ellipse). 11). 20) these unstable modes are linearly independent. 7) follows. 2 4. C o m p a r i s o n s b e t w e e n u p p e r b o u n d s a n d l o w e r b o u n d s . In b o t h cases, space dimension η = 2 or 3, strong solutions to the NavierStokes equations satisfy the energy equation / I I U \ τ at JQ 2 2 dx +ν I Vu [ | dx = f f 2 JQ · udx.

As already remarked before, if a < 0 the periodic solutions occur only at some distance from the equilibrium point (which is unstable). This implies that, when μ is increased past its critical value P j , the family of periodc orbits detaches from the equilibrium point. One can look for periodic solutions for which oc=0 (these are called limit period solutions). The system described by the (approximate) Hamiltonian 1 ^ has then periodic solutions whose minimal period coincides with that of the rotating coordinate system.