D’Alembert

In 1749, D Alembert pointed out that there is a connection between the theory of precession and a figure of the earth. As is well known, precession is caused by the fact that the resultant force of attraction due to celestial bodies, such as Sun and Moon, does not pass through the center of the earth. Correspondingly, there are couple of forces, which tend to turn Earth in such way that the plane of an equator would go through an attracting body and produce a precession. If the earth had a spherical form, then due to spherical symmetry the resultant force passes through the center. However, the spheroidal form does not have such symmetry. Points of the equator or polar axes are exceptions, since the resultant force passes through the earth s center. For all other points this condition is not met. Besides, the position of the resultant force depends also on the distribution of a density inside the earth. Let... 

Diderot, Denis and Jean le Rond d Alembert, eds. Encyclopedia of Diderot d Alembert collaborative translation project.. S.v. "Alchemy," by... 

Since the pressure field depends only on the magnitude of the velocity (see Eq. (1-22)) and since the flow field has fore-and-aft symmetry, the modified pressure field forward from the equator of the sphere is the mirror image of that to the rear. This leads to d Alembert s paradox that the net force acting on the sphere is predicted to be zero. This paradox can only be resolved, and nonzero drag obtained, by accounting for the viscosity of the fluid. For in viscid flow, the surface velocity and pressure follow as... 

Richard N. Schwab, Translators Introduction to d Alemberts Preliminary Discourse to the Encyclopaedia of Diderot, (Indianapolis Bobbs-Merrill, 1963), xi. 

In Eq. (32) we exploited the short-hand notation for the d Alembert operator that contains the second derivatives with respect... 

In U(l) electrodynamics in free space, there are only transverse components of the vector potential, so the integral (158) vanishes. It follows that the area integral in Eq. (157) also vanishes, and so the U(l) phase factor cannot be used to describe interferometry. For example, it cannot be used to describe the Sagnac effect. The latter result is consistent with the fact that the Maxwell-Heaviside and d Alembert equations are invariant under T, which generates the clockwise... 

The Maxwell-Heaviside theory of electrodynamics has no explanation for the Sagnac effect [4] because its phase is invariant under 7 as argued already, and because the equations are invariant to rotation in the vacuum. The d Alembert wave equation of U(l) electrodynamics is also 7 -invariant. One of the most telling pieces of evidence against the validity of the U(l) electrodynamics was given experimentally by Pegram [54] who discovered a little known [4] cross-relation between magnetic and electric fields in the vacuum that is denied by Lorentz transformation. 

Whittaker s early work [27,28] is the precursor [4] to twistor theory and is well developed. Whittaker showed that a scalar potential satisfying the Laplace and d Alembert equations is structured in the vacuum, and can be expanded in terms of plane waves. This means that in the vacuum, there are both propagating and standing waves, and electromagnetic waves are not necessarily transverse. In this section, a straightforward application of Whittaker s work is reviewed, leading to the feasibility of interferometry between scalar potentials in the vacuum, and to a trouble-free method of canonical quantization. 

The derivation of Eq. (218) from Eq. (206) follows from local gauge invariance, and it is always possible to apply a local gauge transform to the vector A, the Maxwell vector potential. The ordinary derivative of the d Alembert wave equation is replaced by an 0(3) covariant derivative. The U(l) equivalent of Eq. (218) in quantum-mechanical (operator) form is Eq. (13), and Eq. (212) is the rigorously correct form of the phenomenological Eq. (25). It can be seen that Eq. (212) is richly structured in the vacuum and must be solved numerically. The vacuum currents present in Eq. (218) can be computed from the right-hand side of the wave equation (212), and these vacuum currents follow from local gauge invariance. 

To an excellent approximation, the four Klein-Gordon equations (443) are d Alembert equations, which are locally gauge-invariant. 

The helicity is gauge-invariant. In the Coulomb gauge, it is obvious that A and C satisfy the d Alembert equation, whose solutions can be written in terms of Fourier transforms... 

Note that we have, by definition, yr = c2y " andyl = c2y, showing that the wave equation is satisfied for all traveling wave shapes yr and vy. However, the derivation of the wave equation itself assumes the string slope is much less than 1 at all times and positions [Morse, 1981]. The traveling-wave solution of the wave equation was first published by d Alembert in 1747 [Lindsay, 1973]. 

Lindsay, 1973] Lindsay, R. B. (1973). Acoustics Historical and Philosophical Development. Dowden, Hutchinson Ross, Stroudsburg. Contains Investigation of the Curve Formed by a Vibrating String, 1747, by Jean le Rond d Alembert. 

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