Quaternionic solutions

To demonstrate the natural unification of the gravitational and electromagnetic aspects of the quaternion field equation (46a) and its conjugate equation (46b), we follow this procedure. Multiply (46a) on the right with the conjugated quaternion solution q, and the conjugated equation (46b) on the left with q-r... 

Observe that, in principle, it is possible to introduce quaternions in the solution of the free rotational part of a Hamiltonian splitting, although there is no compelling reason to do so, since the rotation matrix is usually a more natural coordinatization in which to describe interbody force laws. 

The solutions of the latter equations are the 10 components of the symmetric second-rank metric tensor g iv. The solutions of the factorized equations (46a) [or (46b) are the 16 components of the quaternion metrical field qp (or q ). We will now see that this 16-component metrical quaternion field, indeed, incorporates the gravitational and the electromagnetic fields in terms of their earlier tensor representations. Gravitation entails 10 of the components in the symmetric second-rank tensor g iv. Electromagnetism entails 6 of the components (the 3 components of the electric field and the three components of the magnetic field), as incorporated in the second-rank antisymmetric tensor Fpv. 

In a letter to P. G. Tait in 1871, J. C. Maxwell said the following about the use of quaternions in the laws of physics the virtue of the 4nions lies not so much as yet in solving hard questions as in enabling us to see the meaning of the question and its solutions, Archives, Cavendish Laboratory, Cambridge Univ. 

In the STA solutions corresponding to the one of Darwin s [21], the use of field H of the Hamilton quaternions brings notable simplifications with respect to the standard presentation and a geometrical clarity, which cannot be reached in the complex formalism. 

Note that this equation has already been considered by Sommerfeld ([54], p. 272), the quaternions being expressed in a complex form, but the solutions presented here differ in the fact that they are directly expressed by means of the vectors (u,v) or (n, —w), and (we recall) only the Legendre polynomials P m, pm+i are gj-Qpioyg, the use of P[ i, P 1 being avoided. 

The field H = Cl+(3,0) of the Hamilton quaternions and the ring 0(3,0) of the Clifford biquaternions are relevant of the general theory of the Clifford algebra C1(.E) = Cl(p, n — p) associated with an euclidean space E = Rp,n p. They correspond to the initial construction of the Clifford algebras. Especially, the field of the Hamilton quaternions plays an important role in the solution of the central potential problem. 

Being given a quaternion x = xq + i x + j X2 with vanishing fourth component, X3 = 0, we want to find all quaternions u such that uu = x. We propose the following solution in two steps ... 

First step Find a particular solution u = v = v = vq + i v + j V2 which has also a vanishing fourth component. Since vv = v2 we may use equation (11), which was developed for the complex square root, also for the square root of a quaternion ... 

Horn BKP. Closed-form solution of absolute orientation using unit quaternions. J Opt Soc Am A 1987 4(4) 629-42. 

We see that the imaginary unit i itself satisfies the first two criteria, but obviously not the last. The solution is that q is one of the quaternion units. Quaternions were developed by Hamilton as a further extension of the complex numbers in a progression ... 

It should be noted that the transformation matrix becomes unbounded for / 7t/2. This is the reason for taking other parameterizations of the rotation matrix if /3 tends towards tt/2. Such a reparameterization introduces discontinuities which can be avoided when using a redundant set of rotation coordinates. One typically uses quaternions often also called Euler parameters. These are four coordinates instead of the three angles and one additional normalizing equation, see Ex. 5.1.10. This normalizing equation describes a property of the motion, a so-called solution invariant. Differential equations with invariants will be discussed in Sec. 5.3. 

In the 4D equation, space and time coordinates are inextricably entangled. Its mathematical solutions are hypercomplex functions, or quaternions, without a commutative algebra. Quaternions are used to describe what is known as spherical rotation, also called the spin function, and the complex rotation known as the Lorentz transformation of special relativity. 

The arbitrary rotation of rigid particles is described in the following equations. In this context, the mathematical concept of quaternions is used to represent the angular displacement of the particle within the three-dimensional space [22]. Apart from a higher computational accuracy and efficiency, the main advantage over ordinary rotation matrices is that the solution of quaternions allows an enhanced stability of the numerical solution as a result of non-existent singularities. 

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