Two different types of precessional angular velocities in the complex-octonion space Zi-Hua Weng∗ School of Physics and Mechanical & Electrical Engineering, Xiamen University, Xiamen 361005, China (Dated: Apr. 21, 2017) The paper aims to apply the complex-octonions to explore the two different types of precessional angular velocities for the several particles in the external electromagnetic and gravitational fields. J. C. Maxwell was the first to use the algebra of quaternions to analyze the physical properties of electromagnetic fields. Nowadays, the scholars introduce the complex-octonions to research the physical quantities of electromagnetic and gravitational fields, including the field strength, field source, linear momentum, angular momentum, torque, and force. Further, one formula can be derived from the octonion torque, calculating the precessional angular velocity generated by the gyroscopic torque. Moreover, when the octonion force is equal to zero, it is able to deduce a few equilibrium equations, such as, the force equilibrium equation and precessional equilibrium equation. From the force equilibrium equation, one can infer the angular velocity of revolution for the particles. Meanwhile, from the precessional equilibrium equation, it is capable of ascertaining the precessional angular velocity induced by the torque derivative for the particles, including the angular velocity of Larmor precession. Especially, some ingredients of the torque derivative are in direct proportion to the external field strengths. The study reveals that the precessional angular velocity induced by the torque derivative is independent of that generated by the torque. And the precessional angular velocity, induced by the torque derivative, is relevant to not only the torque derivative, but also the spatial dimension of precessional velocity. It will be of great benefit to understanding further the precessional angular velocity of the spin angular momentum in the quantum mechanics. PACS numbers: 02.10.De, 04.50.-h, 04.80.Cc, 11.10.Kk, 03.65.-w Keywords: precessional angular velocity, field strength, gyroscope, equilibrium equation, torque, spatial dimensions, octonion.
I.
INTRODUCTION
In the magnetic fields, the precessional angular velocity of a boson is distinct from that of a fermion. Since a long time, this problem has been puzzling and intriguing some scholars. For many years, the scholars endeavor to account for this conundrum, attempting to reveal the discrepancy between the precessional angular velocity of the boson with that of the fermion, trying to deepen our understanding of the spin angular momentum. Until recently, the appearance of the electromagnetic and gravitational fields, described with the complex-octonions, replies partially this puzzle. As one of theoretical applications, the relevant inferences of this complex-octonion field theory can be applied to explore the precessional angular velocity of charged/neutral particles. It is able to account for why the precessional angular velocity of a boson is different from that of a fermion in the magnetic fields, unpuzzling why the precessional motion always revolves around the external magnetic flux density for the charged particle. In 1851, Jean Bernard L´eon Foucault began to research the physical properties of the Earth’s rotation, making use of the Foucault pendulum to validate the rotation of the Earth. Some scholars reckoned that he was credited with naming the ‘gyroscope’. Subsequently, the scholars discovered successively numerous physical phenomena relevant to the gyroscopic motions. According to the types of external physical quantities (Table 1), the precessional motions, in the existing studies, can be separated into two categories, the precession generated by the torque, and the precession induced by the torque derivative (in Section 2). The precession induced by the torque ( torque-induced precession, for short) meets the demand of the torque definition derived from the angular momentum (Appendix A). Meanwhile, the precession induced by the torque derivative obeys the precessional equilibrium equation (in Section 4). Some ingredients of the torque derivative are in direct proportion to the external field strengths. And the precession induced by the external field strength (field-induced precession, for short) is a primary representative of the precession induced by the torque derivative. The torque-induced precession can be applied to explore the axial precession and perihelion precession for the Earth and other planets. Condurache et al. [1] offered a comprehensive study of the motion in a central force field with
∗ Electronic
address:
[email protected].
2 respect to a rotating non-inertial reference frame. Stanovnik [2] compared the precession of a Foucault pendulum to the precession of an ideal elastic pendulum, for which the string force is proportional to string length. Icaza-Herrera et al. [3] offered a Lagrangian approach in the case when the angular velocity is the sidereal angular velocity of the Earth. Bergmann et al. [4] explained how the geometrically understood Foucault pendulum can serve as a prototype for more advanced phenomena in physics known as Berry’s phase or geometric phases. Capistrano et al. [5] investigated the anomalous planets precession in the nearly-Newtonian gravitational regime. Xu et al. [6] derived the analytic solution of a planetary orbit disturbed by the solar gravitational oblateness. Barker et al. [7] studied the turbulent damping of precessional fluid motions, in the simplest local computational model of a giant planet (or star), with and without a weak internal magnetic field. Nyambuya [8] presented an improved version of the azimuthally symmetric theory of gravitation model, which is able to explain the anomalous perihelion precession. Sultana et al. [9] investigated the perihelion shift of planetary motion in conformal Weyl gravity using the metric of the static and spherically symmetric solution. Zhao et al. [10] found the exact solution to geodesic equation in the braneworld black hole spacetime by means of Jacobian elliptic function, obtaining the angle of perihelion precession from the zero point of Jacobian elliptic function for Mercury. Xu [11] analyzed the perihelion precessions of orbit with arbitrary eccentricity, making use of the Laplace-Runge-Lenz vector. Zhang et al. [12] ascribed the twin kilohertz Quasi Periodic Oscillations of X-ray spectra to the pseudo-Newtonian Keplerian frequency and the apogee and perigee precession frequency. Saadat et al. [13] visualized the effect of dark matter on solar system and especially perihelion precession of Earth planet. The field-induced precession, especially the Larmor precession, can be utilized to investigate the physical properties of the nano-particles, crystal lattice, and neutron beam and so forth. Bagryansky et al. [14] studied the singlet-triplet oscillations in spin-correlated radical pairs, at magnetic field strengths low for one radical and high for the other. Home et al. [15] provided a detailed analysis for solving the appropriate Schr¨odinger equation, for a spin-polarized plane wave passing through a spin-rotator containing uniform magnetic field. Janosfalvi et al. [16] described the numerous phenomenological equations used in the study of the behavior of single-domain magnetic nano-particles. Li et al. [17] investigated the Larmor precession of a neutral spinning particle in a magnetic field confined to the region of a one-dimensional rectangular barrier. Guo et al. [18] proposed an experimental method to detect the Larmor precession of a single spin with a spin-polarized tunneling current. Muller et al. [19] probed the dynamics of a single hole spin in a single, electrically tunable self-assembled quantum dot molecule formed by vertically stacking (In,Ga)As quantum dots. Rekveldt et al. [20] presented a method of carrying out high-precision measurements of crystal lattice parameters with Larmor precession. Mizukami et al. [21] investigated the magnetization precessions in the epitaxial films grown on MgO substrates, by means of an all-optical pump-probe method. Rekveldt et al. [22] considered the magnetized foils as the flippers to vary the neutron wavelength. The authors [23] reviewed the techniques situation in which the Larmor precession has been used in the neutron spin-echo and neutron depolarization. Hautmann [24] applied the circularly polarized light to inject partially spin-polarized electrons and holes in bulk germanium via both direct and indirect optical transitions. Bouwman [25] modulated the intensity of a neutron beam using Larmor precession techniques. Making a detailed comparison and analysis of the preceding studies, a few primal problems associated with the precessional angular velocity are found as follows. a) Physical quantity. The force may generate the revolution in the curvilinear motion, while the torque is capable of producing the rotation in the curvilinear motion. Similarly, besides the torque, the external magnetic flux density is able to induce the precessional angular velocity as well. There may be many sorts of physical quantities capable of leading to the precessional angular velocity. b) Precessional motion. The force is different from the torque. So the type of curvilinear motion caused by the force can be considered to be independent of that caused by the torque. Analogously, the sort of precessional angular velocity induced by the external magnetic flux density will be considered to be different from that induced by the torque. Is it possible that there are some different types of precessional motions? c) Orientation of precession. The existing classical mechanics and quantum mechanics both are unable to account for a few conundrums relevant to the orientation of precession. For instance, why does the charged particle often revolve around the external magnetic flux density? In the magnetic fields, why is the precessional angular velocity of a boson different from that of a fermion? Presenting a striking contrast to the above is that it is able to resolve a few problems, derived from the classical mechanics and even quantum mechanics, in the electromagnetic and gravitational fields described with the complexoctonions [26], trying to improve some theoretical explanations, associated with the precessional angular velocity and the orientation of precession, to a certain extent. J. C. Maxwell was the first to apply the algebra of quaternions to describe the physical properties of electromagnetic fields. His research methodology inspired the subsequent scholars to employ the quaternions and octonions to explore the gravitational field, electromagnetic field, Dirac wave equation, general relativity, astrophysical jet [27], and dark matter and so forth. When a part of coordinate values are the complex-numbers, the quaternion and octonion are called as the complex-quaternion and complex-octonion respectively.
3 The algebra of quaternions is able to express effectively the precessional angular velocity. In the rotation of a rigid body with respect to one fixed point, the vector terminology can be applied to describe the Eulerian angles, including the nutation angle, precession angle, and intrinsic rotation angle. Making use of the vector terminology to depict the Eulerian angles of the satellite orbits, a few crucial matrices may go wrong occasionally, that is, all of matrix elements of one crucial matrix are equal to zero simultaneously. But this kind of failure will never happen again, when one utilizes the quaternions to express the Eulerian angles of the satellite orbits. By comparison with the vector terminology, the quaternion has obvious advantages to describe the Eulerian angles. According to the precessional equilibrium equation, the external field strength is able to induce the precessional angular velocity of particles, deducing the angular velocity of Larmor precession and some new predictions. However, the precessional angular velocity, induced by the external field strengths, is distinct from that generated by the gyroscopic torque in the theoretical mechanics. Especially, even if the orbital angular momentum is zero, the external electromagnetic strength is still able to induce a new term of precessional angular velocity and a new term of precessional angular momentum, producing a new ingredient of orbital angular momentum. Essentially, the precessional angular momentum is merely one special orbital angular momentum, in the complex-octonion space [28]. In the electromagnetic and gravitational fields, described with the complex-octonions, the precessional angular velocity possesses several significant characteristics as follows. a) Torque derivative. The torque is able to generate the precessional angular velocity, while the torque derivative (the vector, rather than the force) is capable of inducing the precessional angular velocity as well. Some ingredients of the torque derivative may be in direct proportion to the external field strengths. Apparently, the precessional angular velocity, induced by the torque derivative, is distinct from that generated by the torque. They are two different types of precessional angular velocities. b) Precessional axis. When one component of the torque derivative is capable of playing a major role in the precessional motion, the precessional axis of the particle will orient this constituent of the torque derivative. Especially, when the contribution of external field strength plays a major role in the precessional motion, the particle will revolve around the external field strength. In general, the orientation of precession should be composite and intricate. c) Spatial dimension. The precessional angular velocity, induced by the torque derivative, is related with the curl of linear velocity of the precessional motion. Further the curl is relevant to the spatial dimension of the linear velocity. As a result, the precessional angular velocity is multiple-value, in the complex-quaternion space. Its value depends on the actual spatial dimension of the precessional motion. In the paper, by means of the quaternion operator and the octonion field potential (in Section 2), it is able to define the octonion field strength, field source, linear momentum, angular momentum, torque, and force. When the octonion force is equal to zero, it is capable of deducing the force equilibrium equation and precessional equilibrium equation and so forth. The angular velocity of revolution for particles can be derived from the force equilibrium equation, in the electromagnetic or gravitational field. Meanwhile, the angular velocity of precession for particles will be deduced from the precessional equilibrium equation, in the external electromagnetic or gravitational field. The research reveals that the precessional angular velocity, induced by the torque derivative, is multiple-value, and relevant to the spatial dimension of the precessional motion. TABLE I: In the complex-octonion space, there are two different types of precessional angular velocities, caused by two distinct physical quantities respectively. One obeys the formula of octonion torque, while the other meets the demand of the equation of octonion force. precessional angular velocity torque-induced field-induced
equation/formula W = −v0 (iF/v0 + ♦) ◦ L N = −(iF/v0 + ♦) ◦ W
II.
special case Wi1 = −v0 ∂0 L N1 = 0
OCTONION FORCE
In the complex-octonion space for the electromagnetic and gravitational fields, it is able to define the octonion field potential A , field strength F , field source S , linear momentum P , angular momentum L , torque W , and force N , in chronological sequence.
4 A.
Angular momentum
In the complex-quaternion space Hg for the gravitational field, the basis vector is i j , the radius vector is Rg = ir0 i 0 + Σrk i k , and the velocity is Vg = iv0 i 0 + Σvk i k . The gravitational potential is Ag = ia0 i 0 + Σak i k , the gravitational strength is Fg = f0 i 0 + Σfk i k , and the gravitational source is Sg = is0 i 0 + Σsk i k . Herein r = Σrk i k . v = Σvk i k . a = Σak i k . f = Σfk i k . s = Σsk i k . The quaternion operator is ♦ = ii 0 ∂0 + Σi k ∂k . ∇ = Σi k ∂k , ∂j = ∂/∂rj . r0 = v0 t . v0 is the speed of light, and t is the time. rj , vj , aj , sj , and f0 are all real. fk is the complex-number. i is the imaginary unit. i 0 = 1. j = 0, 1, 2, 3. k = 1, 2, 3. In the complex 2-quaternion (short for the second quaternion) space He for the electromagnetic field, the basis vector is I j , the radius vector is Re = iR0 I 0 + ΣRk I k , and the velocity is Ve = iV0 I 0 + ΣVk I k . The electromagnetic potential is Ae = iA0 I 0 + ΣAk I k , the electromagnetic strength is Fe = F0 I 0 + ΣFk I k , and the electromagnetic source is Se = iS0 I 0 +ΣSk I k . Herein R0 = R0 I 0 . V0 = V0 I 0 . A0 = A0 I 0 . F0 = F0 I 0 . S0 = S0 I 0 . R = ΣRk I k . V = ΣVk I k . A = ΣAk I k . F = ΣFk I k . S = ΣSk I k . He = Hg ◦ I 0 . The symbol ◦ denotes the multiplication of octonion. Rj , Vj , Aj , Sj , and F0 are all. Fk is the complex-number. These two orthogonal complex-quaternion spaces, Hg and He , can be combined together to become one complexoctonion space O . In the complex-octonion space for the electromagnetic and gravitational fields, the octonion field source S can be defined as, µS
= −(iF/v0 + ♦)∗ ◦ F = µg Sg + keg µe Se − (iF/v0 )∗ ◦ F ,
(1)
where µ , µg , µe , and keg are coefficients. µg < 0, and µe > 0. The symbol ∗ stands for the octonion conjugate. In the case for single one particle, a comparison with the classical field theory reveals, Sg = mVg , and Se = qVe . m is the density of inertial mass, while q is the density of electric charge. From the octonion field source S , it is able to define the octonion linear momentum as, P = Pg + keg Pe ,
(2)
where Pg = {µg Sg − (iF/v0 )∗ ◦ F}/µg , Pe = µe Se /µg . Pg = ip0 + p , p = Σpk i k . Pe = iP0 + P , P = ΣPk I k , P0 = P0 I 0 . pj and Pj are all real. In the complex-octonion space O , the octonion angular momentum L can be defined from the linear momentum P and the radius vector, R = Rg + keg Re , and so forth (see Ref.[26]). Further the octonion angular momentum L can be separated into, L = Lg + keg Le . Herein Lg = L10 + iLi1 + L1 . L1 = ΣL1k i k , Li1 = ΣLi1k i k . Le = L20 + iLi2 + L2 . L20 = L20 I 0 . L2 = ΣL2k I k , Li2 = ΣLi2k I k . L1j , Li1k , L2j , and Li2k are all real. B.
Force
In the complex-octonion space O , from the octonion angular momentum L , the octonion torque W can be defined as follows, W = −v0 (iF/v0 + ♦) ◦ L ,
(3)
i i where W = Wg + keg We . Wg = iW10 + W10 + iWi1 + W1 . We = iWi20 + W20 + iWi2 + W2 . W10 is the energy. i −W1 is the torque. W1 is the curl of angular momentum, and W10 is the divergence of angular momentum. W20 includes the divergence of magnetic dipole moment, while W2 covers the curl of magnetic dipole moment and the derivative of electric dipole moment. Wi20 contains the divergence of electric dipole moment, and Wi2 comprises the i i curl of electric dipole moment. W1 = ΣW1k i k , Wi1 = ΣW1k i k . W20 = W20 I 0 , Wi20 = W20 I 0 . W2 = ΣW2k I k , i i i i W2 = ΣW2k I k . W1j , W1k , W2j , and W2k are all real. From the octonion torque W , the octonion force N can be defined as,
N = −(iF/v0 + ♦) ◦ W ,
(4)
i where N = Ng + keg Ne . Ng = iN10 + N10 + iNi1 + N1 . Ne = iNi20 + N20 + iNi2 + N2 . N10 is the power, including i comprises the divergence of torque. N1 is the torque derivative. Ni1 some terms in the mass continuity equation. N10 is the force. N20 covers some terms in the current continuity equation. Ni20 contains the term (∂0 W20 + ∇ · Wi2 ). N2 includes the (−∇ ◦ W20 − ∇ × W2 + ∂0 Wi2 ). Ni2 embodies the term (∂0 W2 + ∇ ◦ Wi20 + ∇ × Wi2 ). N1 = ΣN1k i k , i i i i i Ni1 = ΣN1k i k . N2 = ΣN2k I k , N20 = N20 I 0 . Ni2 = ΣN2k I k , Ni20 = N20 I 0 . N1j , N1k , N2j , and N2k are all real.
5 In the complex-octonion space, when the octonion force is equal to zero, that is, N = 0, it is able to deduce eight equilibrium equations, including the force equilibrium equation, precessional equilibrium equation, mass continuity equation, and current continuity equation and so forth (Table 2). TABLE II: The octonion force equation, N = 0 , can be separated into eight independent equilibrium equations. Especially the following four equilibrium equations possess explicit physical meanings. The last two continuity equations belong to the equilibrium equations essentially, in the complex-octonion space. equilibrium equation force equilibrium equation precessional equilibrium equation mass continuity equation current continuity equation
definition Ni1 = 0 N1 = 0 N10 = 0 N20 = 0
III.
EQUILIBRIUM EQUATIONS
In the complex-octonion space O for the electromagnetic and gravitational fields, there exists the octonion force equation, N = 0, under certain circumstances. This equation can be separated into eight independent equilibrium equations. Some of them possess explicit physical meanings, especially the force equilibrium equation, precessional equilibrium equation, mass continuity equation, and current continuity equation. The last two continuity equations belong to the equilibrium equations essentially, in the complex-octonion space.
A.
Force equilibrium equation
In the complex-quaternion space Hg , from the expansion of Eq.(4), the definition of force, Ni1 , can be written as, Ni1 =
i (W10 g/v0 + g × Wi1 /v0 − W10 b − b × W1 )/v0 2 +keg (E ◦ Wi20 /v0 + E × Wi2 /v0 − B ◦ W20 − B × W2 )/v0 i −(∂0 W1 + ∇W10 + ∇ × Wi1 ) ,
(5)
where B is the magnetic flux density, while E is the electric field intensity. g is the gravitational acceleration. And b is one component of gravitational strength, which is similar to the magnetic flux density. In case the field strength F is comparatively weak, the above will approximate to, Ni1 /kp ≈
2 p0 g/v0 + keg (E ◦ P0 /v0 − B × P) − b × p 2 +L10 (g × b + keg E × B)/(v02 kp ) − ∂0 (pv0 ) − ∇(p0 v0 ) ,
(6)
2 (E◦P0 /v0 −B×P) is the electromagnetic where Wi2 ≈ v×P . (p0 g/v0 ) is the gravity, ∂0 (−pv0 ) is the inertial force, keg 0 0 force. ∇(−p0 v0 ) is the energy gradient. p0 = m v0 , p = mv . m is the density of gravitational mass. kp = (k − 1) is one coefficient, with k being the spatial dimension of the vector r or of the velocity v . When the force is equal to zero, the above will deduce the force equilibrium equation,
Ni1 = 0 ,
(7)
where the force consists of the existing force terms in the gravitational and electromagnetic fields, and some tiny force 2 terms. L10 (keg E × B)/(v02 kp ) is in direct proportion to the electromagnetic momentum. Comparing with the force in 2 the classical electromagnetic theory states that keg = µg /µe < 0. B.
Mass continuity equation
In the complex-quaternion space Hg , from the expansion of Eq.(4), the power, N10 , can be defined as, 2 i N10 = (g · W1 /v0 + b · Wi1 )/v0 + keg (E · W2 /v0 + B · Wi2 )/v0 + (∂0 W10 − ∇ · W1 ) ,
(8)
6 further, when the field strength F is comparatively weak, the above will be reduced into, N10 /kp ≈
∂0 (p0 v0 ) − ∇ · (pv0 ) + L10 (b · b − g · g/v02 )/(v0 kp ) 2 2 +keg L10 (B · B − E · E/v02 )/(v0 kp ) + (g · p + keg E · P)/v0 ,
(9)
i where W10 ≈ kp p0 v0 . W1 ≈ kp pv0 . W2 ≈ kp Pv0 . When the power is equal to zero, it is able to derive the mass continuity equation as follows,
N10 = 0 ,
(10)
2 where the power covers the term, (g · p + keg E · P) , which is capable of translating into the Joule heat. And it has an influence on the mass continuity equation directly. In case there is no field strength, the above can be reduced into the mass continuity equation,
∂0 p0 − ∇ · p = 0 ,
(11)
in the classical field theory. Essentially, the mass continuity equation is one type of equilibrium equation, from another point of view.
C.
Current continuity equation
In the complex-quaternion space Hg , from the expansion of Eq.(4), the term, N20 , can be defined as, N20 = (g · W2 /v0 + b · Wi2 )/v0 + (∂0 Wi20 − ∇ · W2 ) + (E · W1 /v0 + B · Wi1 )/v0 ,
(12)
further, when the field strength F is comparatively weak, the above will be simplified into, N20 /kp ≈ ∂0 (P0 v0 ) − ∇ · (Pv0 ) + (b · b + B · B)L20 /(v0 kp ) + g · P/v0 + E · p/v0 ,
(13)
where each component of the field strength F makes a contribution to the above. When the term N20 is equal to zero, it is able to achieve the current continuity equation as, N20 = 0 ,
(14)
where the above includes the interacting term, (g · P + E · p)/v0 , between the gravitational field and electromagnetic field. And the field strength F and linear momentum P and so forth will impact the current continuity equation. In case there is no field strength, the above can be degenerated into the current continuity equation in the classical field theory, ∂0 P0 − ∇ · P = 0 ,
(15)
also the current continuity equation is essentially one type of equilibrium equation. Besides the above three equilibrium equations, there exist other types of equilibrium equations also, especially the precessional equilibrium equation in the next section. From the same octonion equation, N = 0 , it is capable of deducing simultaneously the force equilibrium equation, mass continuity equation, current continuity equation, and precessional equilibrium equation. In case the first three equilibrium equations are all tenable, the last one must be tenable as well. It means that the precessional equilibrium equation has a solid theoretical foundation, in the complex-octonion space. TABLE III: The field-induced precession is a primary representative of the precession induced by the torque derivative. And the precessional angular velocity, induced by the external field strength, is relevant to the spatial dimension of precessional motion and the direction of field strength and so forth. field strength B E b g
precessional angular velocity − → ω p(k) = qB ◦ I 0 /(mk) − → 2 ω p(k) = keg E ◦ W20 /(kp mkv03 ) − → ω p(k) = b/k − → ω = W g/(k mkv 3 ) p(k)
10
p
0
orientation of precession B ◦ I0 E ◦ I0 b g
7 IV.
PRECESSIONAL EQUILIBRIUM EQUATION
In the complex-octonion space O , when the octonion force N is equal to zero, it is capable of inferring eight independent equilibrium equations, especially the equilibrium equation, N1 = 0 . Further, it is able to deduce the angular velocities of precession for the charged/neutral particles, from the equilibrium equation, N1 = 0 . This equilibrium equation is called as the precessional equilibrium equation temporarily. The equation will be disturbed by the gravitational strength, electromagnetic strength, torque, and spatial dimension and so forth. The precessional equilibrium equation, N1 = 0 , can be expanded into, i (W10 g/v0 + g × W1 /v0 + W10 b + b × Wi1 )/v0 + (∂0 Wi1 − ∇W10 − ∇ × W1 )
0=
2 +keg (E ◦ W20 /v0 + E × W2 /v0 + B ◦ Wi20 + B × Wi2 )/v0 ,
(16)
where the term in the above is the torque-derivative term, rather than the force term, although both of which are the vectors and possess the same dimension. Obviously the field strength and so forth may exert an influence on the precessional equilibrium equation (Table 3). In the complex-octonion space, the above can be reduced into a few special cases as follows. A.
Torque-derivative term
In case two terms, ∂0 Wi1 and ∇ × W1 , play a major role in the precessional motion, meanwhile there is no field strength, and other tiny terms can be neglected, the precessional equilibrium equation, N1 = 0 , can be degenerated into, ∂0 Wi1 − ∇ × W1 = 0 ,
(17) ∂0 Wi1
, is relevant to some influence factors, according to the definition of torque. where the torque-derivative term, ∇ × W1 ≈ kp mv0 ∇ × v . The linear velocity for the objects consists of the linear velocity caused by the motions of revolution, rotation, and precession. In the paper, we discuss merely the linear velocity vp caused by the precessional − − motion. As a result, ∇ × vp = k → ωp . → ω p is the angular velocity of precession, while k is the spatial dimension of linear velocity vp . Further the above will be simplified into − ∂ Wi − k mkv → ω =0, (18) 0
p
1
0
p
where the angular velocity of precession is related with the spatial dimension, mass, and torque-derivative term and so forth. The above states that the torque-derivative term, ∂0 Wi1 , will produce the angular velocity of precession, when there is no field strength. B.
Magnetic field
2 If two terms, keg B ◦ Wi20 /v0 and ∇ × W1 , play a major role in the precessional motion, meanwhile there is no electric field intensity nor the gravitational strength, and other tiny terms can be neglected, the precessional equilibrium equation, N1 = 0 , can be simplified into, 2 keg B ◦ Wi20 /v0 − ∇ × W1 = 0 , i 2 where the term Wi20 is similar to the energy W10 . Wi20 ≈ kp P0 v0 . keg P0 = qV0 I 0 . V0 /v0 ≈ 1. Further the above will be reduced into, − qB ◦ I − km→ ω =0,
(19)
(20) → − where the angular velocity of precession is, ω p(1) = qB ◦ I 0 /m , when k = 1. The angular velocity of precession is, → − − ω p(2) = qB ◦ I 0 /(2m) , when k = 2. And the angular velocity of precession is, → ω p(3) = qB ◦ I 0 /(3m) , when k = 3. According to the multiplication of octonions, the term, B ◦ I 0 , is the vector in the complex-quaternion space Hg . The above means that the magnetic flux density B will induce the angular velocity of precession, revolving around the direction, B ◦ I 0 , for the charged object. The inference can be applied to explicate the angular velocity of Larmor precession, for the charged particle and the spin angular momentum in the magnetic field (see Ref.[28]). This will be helpful to understand the spin angular momentum in the quantum mechanics. 0
p
8 C.
Electric field
2 When two terms, keg (E ◦ W20 )/v02 and ∇ × W1 , play a major role in the precessional motion, meanwhile there is no magnetic flux density nor the gravitational strength, and other tiny terms can be neglected, the precessional equilibrium equation, N1 = 0 , can be degraded into, 2 keg E ◦ W20 /v02 − ∇ × W1 = 0 ,
(21)
where the term W20 is relevant to the field strength and angular momentum and so forth. The term, E ◦ I 0 , is the vector in the complex-quaternion space Hg , according to the multiplication of octonions. The above reveals that the electric field intensity E will generate the angular velocity of precession, with the orientation of precession, E ◦ I 0 , for the charged object. The term W20 is more intricate than the term Wi20 , so the angular velocity of precession in the electric field intensity E will be much more complicated than that in the magnetic flux density B . And it can be utilized to unpuzzle some precessional phenomena of charged particles in the electric field.
D.
Component of gravitational strength
i When two terms, W10 b/v0 and ∇ × W1 , play a major role in the precessional motion, meanwhile there is no electromagnetic strength nor the gravitational acceleration, and other tiny terms can be neglected, the precessional equilibrium equation, N1 = 0 , can be degenerated into, i W10 b/v0 − ∇ × W1 = 0 ,
(22)
i where the energy W10 includes the work, kinetic energy, potential energy, and proper energy and so forth. The above can be further simplified into,
− b − k→ ωp = 0 ,
(23)
− − where the angular velocity of precession is, → ω p(1) = b , when k = 1. The angular velocity of precession is, → ω p(2) = b/2, → − when k = 2. And the angular velocity of precession is, ω p(3) = b/3 , when k = 3. The above states that the component b of gravitational strength will induce the angular velocity of precession, with the orientation of precession, b , for the neutral object. And it can be applied to account for the dynamic properties of the astrophysical jets (see Ref.[27]), including the precession, rotation, and collimation and so forth.
E.
Gravitational acceleration
In case two terms, W10 g/v02 and ∇ × W1 , play a major role in the precessional motion, meanwhile there is no electromagnetic strength nor the component b of gravitational strength, and other tiny terms can be neglected, the precessional equilibrium equation, N1 = 0 , can be degraded into, W10 g/v02 − ∇ × W1 = 0 ,
(24)
where the term W10 is relevant to the field strength and angular momentum and so forth. The above states that the gravitational acceleration will result in the angular velocity of precession, with the i orientation of precession, g , for the neutral object. The term W10 is more involved than the term W10 , so the angular velocity of precession in the gravitational acceleration g will be much more complicated than that in the component b of gravitational strength. And it can be applied to explain some precessional phenomena of neutral particles in the gravitational field g . In the complex-octonion space, the torque and force may generate different types of curvilinear motions, for instance, the rotation and revolution. Similarly, the torque and torque derivative can induce different types of precessional angular velocities, such as, the gyroscopic precession and field-induced precession. In the section, it should be noted that the precessional angular velocity, induced by the torque derivative, is quite a contrast to that derived from the octonion torque, Eq.(3).
9 V.
PRECESSIONAL ANGULAR VELOCITY
In the complex-octonion space O , some factors of the torque derivative may exert an influence on the precessional angular velocity, including the torque, partial derivative of torque, gravitational strength, and electromagnetic strength and so on. The influences of these factors on the precessional motions may be different from each other. So it is necessary to contrast and analyze various properties of the precessional angular velocities, especially the orientation of precession, relevant equations, and spatial dimension and so forth.
A.
Orientation of precession
From Eq.(18), it is found that the partial derivative of torque, with respect to time, is able to induce directly the precessional angular velocity. If the torque derivative term, v0 ∂0 Wi1 , is periodic, the precessional angular velocity, → − ω p , will be periodic as well. And the period of precessional angular velocity is identical with that of the torque derivative term. Meanwhile the direction of precessional angular velocity orients the torque derivative term. For the electromagnetic fields, either of two components, B and E , is capable of inducing the precessional angular − velocity directly. From Eq.(20), it is found that the precessional angular velocity is, → ω p(k) = qB ◦ I 0 /(mk) , in the magnetic flux density B . And the direction of precessional angular velocity orients the vector B ◦ I 0 . From Eq.(21), the precessional angular velocity is comparatively complicated, and relevant to various factors, in the electric field intensity E . And the direction of precessional angular velocity is along the vector E ◦ I 0 . In terms of the gravitational fields, either of two constituents, g and b , is able to induce the precessional angular − velocity directly. According to Eq.(23), it is found that the precessional angular velocity is, → ω p(k) = b/k , in the component b of gravitational strength. And the direction of precessional angular velocity orients the vector b . From Eq.(24), the precessional angular velocity is also comparatively complicated, and relevant to assorted factors, in the gravitational acceleration g . And the direction of precessional angular velocity is along the vector g . Besides these factors in the above, certain other factors in Eq.(16) can directly lead to the precessional angular velocities as well.
B.
Spatial dimension
According to Eq.(17), the precessional angular velocity is relevant to the curl of linear velocity of the precessional motion, while the curl is relevant to the spatial dimension k of the linear velocity. In the complex-quaternion space Hg , the possible value of spatial dimension is, k = 1, 2, 3. Consequently, the precessional angular velocity may possess three types of possible values. In terms of the charged particles in the magnetic field B , the precessional angular velocity occupies three types of possible values, from Eq.(20). When k = 1, the precessional angular velocity corresponds to the one-dimensional precessional motion. It is possible to speculate that the precessional motion of Boson particles is one-dimensional. If k = 2, the precessional angular velocity may correspond to the two-dimensional precessional motion. We can speculate that the precessional motion of Fermion particles is two-dimensional. In case k = 3, the precessional angular velocity must correspond to the three-dimensional precessional motion of some particles. It is interesting to speculate that the possible values of the precessional angular velocities should be altered, in case the Bosons or Fermions and other particles were compelled to experience the three-dimensional precessional motions, in the magnetic fields. Similarly, the precessional angular velocity has three types of possible values, for the neutral particles in the gravitational field b . In case the precessional motion of neutral particles is one-dimensional, the value of the precessional angular velocity corresponds to the case k = 1. When the precessional motion of neutral particles is two-dimensional, the value of the precessional angular velocity may correspond to the case k = 2. If the precessional motion of some neutral particles is three-dimensional, the value of the precessional angular velocity will correspond to the case k = 3. In other words, the spatial dimension of precessional motion exerts an influence on the possible values of precessional angular velocity, for the neutral particles in the gravitational field b . In the electric field E and gravitational field g , there exist three possible values of precessional angular velocities as well. Furthermore, from Eq.(16), the spatial dimension of precessional motion of particles will also make a contribution to the possible values of precessional angular velocities, caused by the other physical quantities, besides the above field strengths.
10 C.
Two equations
In the curvilinear motions, the linear velocity of an object consists of the linear-velocity terms caused by the revolution, rotation, and precession. Apparently, the precessional motion is different from the revolution motion or rotation motion. And the necessary equation for precessional motion is distinct from that for the revolution motion or rotation motion. The force and torque can generate the revolution motion and rotation motion respectively. The force is different from the torque. So the angular velocity of revolution, caused by the force, is considered to be independent of the angular velocity of rotation, caused by the torque. Both of which may be dissimilar in other physical properties either. For instance, these two types of angular velocities may consume (or store) different energies. Similarly, the torque derivative is distinct from the torque. As a result, the angular velocity of precession induced by the torque derivative is absolutely different from that induced by the torque. They are two disparate types of angular velocities of precession. One obeys the octonion torque formula, Eq.(3), while the other meets the octonion force formula, Eq.(4). In terms of other physical properties, both of which may be diverse either.
VI.
EXPERIMENT PROPOSAL
In the complex-octonion space O , the influence of two constituents, g and b , of gravitational strength on the − precessional angular velocity, → ω p , is comparatively tough to be validated within the laboratory experiments at present. Only the influence of the constituent, b , of gravitational strength may be observed in the astrophysical phenomena, especially the explanation of the astrophysical jets. Fortunately, it is able to validate directly the influence of the − torque-derivative term, v0 ∂0 Wi1 , and electromagnetic strength (E and B) on the precessional angular velocity, → ω p , in the laboratory. For instance, it is capable of measuring the influence of the torque-derivative term on the precessional angular velocity, for the rotating rotors. In the super-strong magnetic field (or electric field), it is able to explore the influencing degree of the external fields on the precessional angular velocity, in terms of the suspended and tiny charged objects. a) Time-varying term. From Eq.(18), it is found that the precessional direction of the rotating rotor will orient the torque-derivative term, v0 ∂0 Wi1 , if the external torque, −Wi1 , is a time-varying term. In other words, when we accelerate or decelerate the rotation motion of rotors, it must generate the precessional motion. No matter the direction of torque-derivative term, v0 ∂0 Wi1 , is oriented or reversed to that of the rotation motion of rotors, it is still able to produce the precessional motion. Consequently, on the basis of existing gyroscopic experiments, it is feasible to validate the influence of the torque-derivative term on the precessional angular velocity in the experimental technique, after improving appropriately the existing experimental facilities. b) Electromagnetic field. According to Eq.(20), when there is the external magnetic field B , the precessional direction of the rotating charged object will orient the vector, B ◦ I 0 . Even if the angular velocity of rotation is equal to zero, the external magnetic field is still able to induce the precessional angular velocity for the charged objects. Further, the spatial dimension of linear velocity will make a contribution to the precessional angular velocity of charged objects, especially the precessional motions in the three-dimensional magnetic fields. In the Electron Spin Resonance experiments to study the sample materials (such as, Di-Phehcryl Pircyl Hydrazal) with unpaired electrons, when we apply the Crystal Lattice Vibration methods (for instance, neutron inelastic scattering) to vibrate the crystal lattices, it is able to result in the three-dimensional precessional motions of unpaired electrons in the sample materials, measuring the absorption spectrum of electromagnetic waves caused by the three-dimensional precessional motions. In a similar way, from Eq.(21), when the external three-dimensional electric field E is applied to the suspended and tiny charged objects, we can also detect the spectrum of electromagnetic waves, caused by the precessional motions. Certainly, the spectrum of electromagnetic waves in the external electric fields may be much more intricate than that in the external magnetic fields. c) Gravitational field. When there exists the external gravitational strength g , the precessional direction of the rotating neutral object will orient the vector, g , from Eq.(24). No matter the direction of external gravitational strength g , is oriented or reversed to that of the rotation motion of rotating neutral object, it is still able to induce the precessional motion. Even if the angular velocity of rotation is equal to zero, the external gravitational strength is still able to induce the precessional angular velocity for the neutral objects. Further, the spatial dimension of linear velocity will make a contribution to the precessional angular velocity of neutral objects, especially the precessional motions in the three-dimensional gravitational field, g . Therefore, when we apply the external three-dimensional gravitational field, g , to the tiny neutral objects, it is capable of measuring the precessional motions, although this precessional angular velocity is difficult to detect. The validation of the above experiment proposal will be of great benefit to investigating the further features of precessional angular velocities.
11 VII.
CONCLUSIONS AND DISCUSSIONS
The complex-quaternion space can be applied to study the physical properties of electromagnetic and gravitational fields. And the complex-octonion space can be utilized to explore the physical quantities of electromagnetic and gravitational fields simultaneously, including the field potential, field strength, field source, linear momentum, angular momentum, torque, and force. When the octonion force is equal to zero, it is able to deduce eight independent equilibrium equations, including the force equilibrium equation, precessional equilibrium equation, mass continuity equation, and current continuity equation. Essentially, the two continuity equations belong to the equilibrium equations. Moreover, the eight equilibrium equations are relevant to the spatial dimension of linear velocity. From the precessional equilibrium equation, it is found that the torque-derivative term, gravitational strength, and electromagnetic strength have an influence on the precessional angular velocities of neutral/charged objects. Especially, the magnetic flux density B will make a contribution on the precessional angular velocities of charged objects, explaining the angular velocity of Larmor precession, for the spin angular momentum. And the component b of gravitational strength is able to impact the precessional angular velocities of neutral objects, accounting for the precessional phenomena of astrophysical jets. Either of torque and torque derivative may exert an influence on the angular velocity of precession. Especially, the external field is able to make a contribution on the precessional angular velocities of neutral/charged objects. Obviously, this precessional angular velocity is distinct from that caused by the conventional gyroscopic torque in the theoretical mechanics. The former obeys the octonion force formula, while the latter meets the requirement of the octonion torque. It should be noted that the paper discussed merely some simple cases about the influence of torque derivative on the angular velocities of precession. However it is clearly states that the external field strengths and the curl of linear velocity both are capable of inducing the precessional angular velocities of neutral/charged objects, including the magnitude and direction. In the following study, we will explore theoretically certain other influence factors of precessional angular velocities, by means of the precessional equilibrium equation; and validate the influence of the curl of linear velocities on the precessional angular velocity in the three-dimensional magnetic fields, making use of some improved experiment devices; and research further the physical properties of spin angular momentum in the quantum mechanics, on the basis of the precessional angular velocity induced by the torque derivative.
Acknowledgments
This project was supported partially by the National Natural Science Foundation of China under grant number 60677039.
APPENDIX A: GYROSCOPIC TORQUE
In the complex-octonion space, the octonion torque formula can be degenerated into the torque formula (see Ref.[26]). Later, in the complex-quaternion space, the torque formula is able to deduce the precessional angular velocity caused by the conventional gyroscopic torque. In the complex-octonion space, the octonion torque is, W = −v0 (iF/v0 + ♦) ◦ L ,
(A1)
further, in case there is no field strength, that is, F = 0 , the above will be reduced into, W = −v0 ♦ ◦ L .
(A2)
In the complex-quaternion space, the above will be simplified into, Wi1 = −v0 ∂0 L ,
(A3)
where −Wi1 is conventional torque, and is the major ingredient of W . And L is the orbital angular momentum, and is the primary constituent of L . The above is the definition of torque in the theoretical mechanics. Subsequently, according to the above, the conventional gyroscopic torque can produce the precessional angular velocity. Apparently, the above is independent of the octonion force, N = −(iF/v0 + ♦) ◦ W ,
(A4)
12 in the complex-octonion space. When N = 0, the above will be separated into eight independent equilibrium equations, including the force equilibrium equation and the precessional equilibrium equation. According to the precessional equilibrium equation, the torque derivative will induce another type of precessional angular velocity, which is distinct from that caused by the gyroscopic torque.
[1] D. Condurache and V. Martinusi, Foucault Pendulum-Like Problems: A Tensorial Approach, International Journal of Non-Linear Mechanics, Vol. 43, No. 8, 743–760, 2008. [2] A. Stanovnik, The Foucault pendulum with an ideal elastic suspension string, European Journal of Physics, Vol. 27, No. 2, 205–213, 2006. [3] M. de Icaza-Herrera, V. M. Castano, The Foucault pendulum viewed as a spherical pendulum placed in a rotating reference system, Canadian Journal of Physics, Vol. 94, No. 1, 15–22, 2016. [4] J. von Bergmann, H. von Bergmann, Foucault pendulum through basic geometry, American Journal of Physics, Vol. 75, No. 10, 888–892, 2007. [5] A. J. S. Capistrano, J. A. M. Penagos, M. S. Alarcon, Anomalous precession of planets on a Weyl conformastatic solution, Monthly Notices of the Royal Astronomical Society, Vol. 463, No. 2, 1587–1591, 2016. [6] Y. Xu, Y.-X. Yang, Q. Zhang, G.-C. Xu, Solar oblateness and Mercury’s perihelion precession, Monthly Notices of the Royal Astronomical Society, Vol. 415, No. 4, 3335–3343, 2011. [7] A. J. Barker, On turbulence driven by axial precession and tidal evolution of the spin-orbit angle of close-in giant planets, Monthly Notices of the Royal Astronomical Society, Vol. 460, No. 3, 2339–2350, 2016. [8] G. G. Nyambuya, Azimuthally symmetric theory of gravitation - II. On the perihelion precession of solar planetary orbits, Monthly Notices of the Royal Astronomical Society, Vol. 451, No. 3, 3034–3043, 2015. [9] J. Sultana, D. Kazanas, J. L. Said, Conformal Weyl gravity and perihelion precession, Physical Review D, Vol. 86, No. 8, id. 084008, 2012. [10] F. Zhao, F. He, Exact Solution to Geodesic Motion Equation in the Braneworld Black Hole Spacetime and the Perihelion Precession of Mercury, International Journal of Theoretical Physics, Vol. 51, No. 5, 1435–1441, 2012. [11] F. Xu, Perihelion precession from power law central force and magnetic-like force, Physical Review D, Vol. 83, No. 8, id. 084008, 2011. [12] C. M. Zhang, A. Dolgov, Quasi Periodic Oscillations in X-ray Neutron Star and Probes of Perihelion Precession of General Relativity, International Journal of Modern Physics D, Vol. 11, No. 4, 503–510, 2002. [13] H. Saadat, S. N. Mousavi, M. Saadat, N. Saadat, A. M. Saadat, The Effect of Dark Matter on Solar System and Perihelion Precession of Earth Planet, International Journal of Theoretical Physics, Vol. 49, No. 10, 2506–2511, 2010. [14] V. A. Bagryansky, V. I. Borovkov, Yu. N. Molin, Singlet-triplet oscillations of spin-correlated radical pairs due to the Larmor precession in low magnetic fields, Molecular Physics, Vol. 100, No. 8, 1071–1078, 2002. [15] D. Home, A. K. Pan, A. Banerjee, Reexamining Larmor precession in a spin-rotator: testable correction and its ramifications, European Physical Journal D, Vol. 67, No. 4, id. 72, 2013. [16] Zs. Janosfalvi, J. Hakl, P. F. de Chatel, Larmor precession and Debye relaxation of single-domain magnetic nanoparticles, Advances in Condensed Matter Physics, Vol. 2014, id. 125454, 2014. [17] Z.-J. Li, J.-Q. Liang, D. H. Kobe, Larmor precession and barrier tunneling time of a neutral spinning particle, Physical Review A, Vol. 64, No. 4, id. 042112, 2001. [18] X.-D. Guo, L. Dong, Y. Guo, X.-Y. Shan, J.-M. Zhao, X.-H. Lu, Detecting Larmor Precession of a Single Spin with a Spin-Polarized Tunneling Current, Chinese Physics Letters, Vol. 30, No. 1, id. 017601, 2013. [19] K. Muller, A. Bechtold, C. Ruppert, C. Hautmann, J. S. Wildmann, T. Kaldewey, M. Bichler, H. J. Krenner, G. Abstreiter, M. Betz, J. J. Finley, High-fidelity optical preparation and coherent Larmor precession of a single hole in an (In,Ga)As quantum dot molecule, Physical Review B, Vol. 85, No. 24, id. 241306, 2012. [20] M. T. Rekveldt, T. Keller, R. Golub, Larmor precession, a technique for high-sensitivity neutron diffraction, Europhysics Letters, Vol. 54, No. 3, 342–346, 2001. [21] S. Mizukami, S. Iihama, Y. Sasaki, A. Sugihara, R. Ranjbar, and K. Z. Suzuki, Magnetic relaxation for Mn-based compounds exhibiting the Larmor precession at THz wave range frequencies, Journal of Applied Physics, Vol. 120, No. 14, id. 142102, 2016. [22] M. Th. Rekveldt, W. G. Bouwman, W. H. Kraan, T. V. Krouglov, J. Plomp, Larmor precession applications: magnetised foils as spin flippers in spin-echo SANS with varying wavelength, Physica B: Physics of Condensed Matter, Vol. 335, No. 1-4, 164–168, 2003. [23] M. T. Rekveldt, W. G. Bouwman, W. H. Kraan, S. V. Grigoriev, O. Uca, T. Keller, Overview of new Larmor precession techniques, Applied Physics A: Materials Science & Processing, Vol. 74, No. s01, 323–325, 2002. [24] C. Hautmann, B. Surrer, M. Betz, Ultrafast optical orientation and coherent Larmor precession of electron and hole spins in bulk germanium, Physical Review B, Vol. 83, No. 16, id. 161203, 2011. [25] W. G. Bouwman, C. P. Duif, R. Gahler, Spatial modulation of a neutron beam by Larmor precession, Physica B: Physics of Condensed Matter, Vol. 404, No. 17, 2585–2589, 2009. [26] Z.-H. Weng, Angular momentum and torque described with the complex octonion, AIP Advances, Vol. 4, No. 8, id. 087103,
13 2014. [Erratum: ibid. Vol. 5, No. 10, id. 109901, 2015] [27] Z.-H. Weng, Dynamic of astrophysical jets in the complex octonion space, International Journal of Modern Physics D, Vol. 24, No. 10, id. 1550072, 2015. [28] Z.-H. Weng, Spin Angular Momentum and Proton Spin Decomposition in Complex Octonion Spaces, International Journal of Geometric Methods in Modern Physics, Vol. 14, No. 6, id. 1750102, 2017.
