By H. K. Moffatt (ed.), G. M. Zaslavsky (ed.), P. Comte (ed.), M. Tabor (ed.)

The intimate courting among 3-dimensional vortex dynamics and topology was once well-known as early as 1869 through Lord Kelvin, whose discoveries in fluid dynamics resulted in the improvement of knot idea as a well-defined department of topology. it's only within the final 25 years, even though, with the parallel stimulus of the advance of magnetohydrodynamics in astrophysical and geophysical contexts, that the good capability of topological options in fluid and plasma has been absolutely well-known. This quantity presents a entire survey of this interdisciplinary box. The proper heritage in knot conception, circulation kinematics, dynamo concept, and leisure below topological constraints, is supplied by way of the introductory chapters of half I. those topics are built within the next papers that are grouped less than the subsequent headings: half II: leisure and minimal strength States; half III: Helicity, Linkage, and move Topology; half IV: The Euler Equations: Extremal homes and Finite-Time Singularities; half V: Vortex Interactions and the constitution of Turbulence; half VI: Chaos, Instability, and Dynamo conception

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C) By variant calculations specifically utilizing ENCKE'S method for integrating special perturbations-with "engineering perturbations" (as mentioned above) instead of astronomical ones. These integrations lead to differences directly proportional to the partial differential coefficients, rather than gross values that must be differenced, as in COWELL'S method. (d) By the "adjoint method", in which linearized variants are integrated backwards to give the partial differential coefficients directly.

SHUTE of the Theoretical Division has applied the HALPHENGORIACHEV method extensively to the investigation of long range stability of artificial satellites and compared the results for some satellites with numerical integration using ENcKE's method as programmed by the Republic Aviation Corporation, Farmingdale, New York. Figures 2 and 3 show the results obtained using both methods for the hypothetical satellite IMP ("Interplanetary Monitoring Probe"). 95 initially. The comparison clearly indicates a relative unimportance of short period terms and of their long range cross actions over an extended interval of time.

For such orbits the mean motion of g is negative. Hence the semi-major axis can be chosen such that the mean motion of g + A' is approximately zero. A simplifying circumstance for polar orbits is that the mean motion of h is zero for 1 = 90°. Consequently the last three terms ofLlF have arguments with approximately zero mean motion. The first three terms can be eliminated in the same manner as the periodic oblateness terms, and the principal features of the motion in the resonance region in the vicinity of 1 = 90° can be obtained by limiting LlF to the last three terms.