By Felix Klein
The idea of the pinnacle. quantity II. improvement of the speculation on the subject of the Heavy Symmetric Top is the second one in a sequence of 4 self-contained English translations of the vintage and definitive remedy of inflexible physique motion.
* entire and unabridged presentation with fresh advances and
* Annotations via the translators offer insights into the character of
technological know-how and arithmetic within the overdue nineteenth century
* every one quantity interweaves conception and applications
the 1st quantity verified the final kinematic and kinetic foundations of the idea. quantity II discusses the movement of the symmetric best with a set aid element, below the impression of gravity, in all its info. The concept of the Top was once initially awarded through Felix Klein as an 1895 lecture at Göttingen college that was once broadened in scope and clarified because of collaboration with Arnold Sommerfeld.
Graduate scholars and researchers drawn to theoretical and utilized mechanics will locate this a radical and insightful account.
different works during this sequence contain Volume I. advent to the Kinematics and Kinetics of the Top, Volume III. Perturbations. Astronomical and Geophysical Applications, and Volume IV. Technical functions of the speculation of the Top.
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Extra info for The Theory of the Top. Volume II: Development of the Theory in the Case of the Heavy Symmetric Top
The points in the previous ﬁgure at which the rotation sense about the vertical changes now come together in the north pole of the sphere. To envision the conﬁguration of the trajectories with progressively increasing n, we ﬁnally draw upon the limiting case n = ∞, for which, according to page 202, the motion again becomes a regular precession. While the inner parallel circle is reduced to a point in the case n = N , it coincides in the case n = ∞ with the exterior bounding circle. We will again presume, on the basis of our continuity principle, that the parallel circle in question continuously widens with increasing values of n (N < n < ∞).
Relation between the motions of diﬀerent tops. 233 of all those symmetric tops with the same initial impulse, the same initial position of the ﬁgure axis, the same gravitational turning-moment, and the same equatorial principal moment of inertia. In addition to the impulse curves, the trajectories of the apices of the tops will also be identical for all such motions. We can also, however, give an immediate geometric account of this characteristic coincidence between the motions of the diﬀerent tops.
We now investigate the elliptic integral constructed for t, in its dependence on u, in somewhat more detail. Here we use the evident fact that in mechanics time signiﬁes not only a real, but also a continuously increasing quantity, and that dt must therefore be necessarily positive. As the lower limit of the integral, we take the smaller of the two roots of U = 0 between −1 and +1, which we denote (as in the ﬁgure) by e. Starting from this point, we must let u occur only in the mechanically useful interval between e and e , and indeed, according to the previously established principle, in the upper covering of this interval, so that we obtain for du dt = √ U a positive value.
The Theory of the Top. Volume II: Development of the Theory in the Case of the Heavy Symmetric Top by Felix Klein