Papers by Daniel McNamara
We studied the orbit of an electron revolving around an infinitely massive nucleus of a large cla... more We studied the orbit of an electron revolving around an infinitely massive nucleus of a large classical Hydrogen atom subject to an AC electric field oscillating perpendicular to the electron's circular orbit. Using perturbation theory in geometric algebra, we show that the equation of motion of the electron perpendicular to the unperturbed orbital plane satisfies a forced simple harmonic oscillator equation found in Lorentz dispersion law in Optics. We show that even though we did not introduce a damping term, the initial orbital position and velocity of the electron results to a solution whose absorbed energies are finite at the dominant resonant frequency $\omega=\omega_0$; the electron slowly increases its amplitude of oscillation until it becomes ionized. We computed the average power absorbed by the electron both at the perturbing frequency and at the electron's orbital frequency. We graphed the trace of the angular momentum vector at different frequencies. We showed t...
Applied Optics, 2015
We used Geometric Algebra to compute the paths of skew rays in a cylindrical, step-index multimod... more We used Geometric Algebra to compute the paths of skew rays in a cylindrical, step-index multimode optical fiber. To do this, we used the vector addition form for the law of propagation, the exponential of an imaginary vector form for the law of refraction, and the juxtaposed vector product form for the law of reflection. In particular, the exponential forms of the vector rotations enables us to take advantage of the addition or subtraction of exponential arguments of two rotated vectors in the derivation of the ray tracing invariants in cylindrical and spherical coordinates. We showed that the light rays inside the optical fiber trace a polygonal helical path characterized by three invariants that relate successive reflections inside the fiber: the ray path distance, the difference in axial distances, and the difference in the azimuthal angles. We also rederived the known generalized formula for the numerical aperture for skew rays, which simplifies to the standard form for meridional rays.
Physics and Chemistry of the Earth, Parts A/B/C, 2009
has obtained geomagnetic data in the Circum-pan Pacific Magnetometer Network (CPMN) region for ov... more has obtained geomagnetic data in the Circum-pan Pacific Magnetometer Network (CPMN) region for over 10 years, and has recently deployed a new real-time Magnetic Data Acquisition System (MAGDAS) in the CPMN region and an FM-CW radar network along the 210°magnetic meridian (MM) for space weather research and applications. This project intends to get the MAGDAS network fully operational and provide data for studies on space and lithosphere weather. In connection with this project, we propose a new ultra-low frequency (ULF) wave analysis method to study ULF anomalies associated with large earthquakes using magnetic data. From a case study of the 1999/05/12 Kushiro earthquake with magnitude M = 6.4, we found a peculiar increase of H-component power ratio A R /A M of Pc 3 magnetic pulsations a few weeks before the earthquake, where A R is the power obtained at Rikubetsu station (r = 61 km) near the epicenter and A M is the power obtained at a remote reference station, Moshiri (r = 205 km). It is also found that the H-component power ratio A D / A Y of Pc 3 increased three times just a few weeks before the earthquake and after one week decreased to the normal level, where A D is one-day power at Rikubetsu station and A Y is the one-year-average power.
Journal of Geophysical Research: Space Physics, 2011
An empirical model of the quiet daily geomagnetic field variation has been constructed based on g... more An empirical model of the quiet daily geomagnetic field variation has been constructed based on geomagnetic data obtained from 21 stations along the 210 Magnetic Meridian of the Circum-pan Pacific Magnetometer Network (CPMN) from 1996 to 2007. Using the least squares fitting method for geomagnetically quiet days (Kp ≤ 2+), the quiet daily geomagnetic field variation at each station was described as a function of solar activity SA, day of year DOY, lunar age LA, and local time LT. After interpolation in latitude, the model can describe solar-activity dependence and seasonal dependence of solar quiet daily variations (S) and lunar quiet daily variations (L). We performed a spherical harmonic analysis (SHA) on these S and L variations to examine average characteristics of the equivalent external current systems. We found three particularly noteworthy results. First, the total current intensity of the S current system is largely controlled by solar activity while its focus position is not significantly affected by solar activity. Second, we found that seasonal variations of the S current intensity exhibit northsouth asymmetry; the current intensity of the northern vortex shows a prominent annual variation while the southern vortex shows a clear semi-annual variation as well as annual variation. Thirdly, we found that the total intensity of the L current system changes depending on solar activity and season; seasonal variations of the L current intensity show an enhancement during the December solstice, independent of the level of solar activity.
Arxiv preprint arXiv:0812.2979, 2008
We use the vector wedge product in geometric algebra to show that Poisson commutator brackets mea... more We use the vector wedge product in geometric algebra to show that Poisson commutator brackets measure preservation of phase space areas. We also use the vector dot product to define the Poisson anticommutator bracket that measures the preservation of phase space angles. We apply these brackets to the paraxial meridional complex height-angle ray vectors that transform via a 2×2 matrix, and we show that this transformation preserves areas but not angles in phase space. The Poisson brackets here are expressed in terms of the coefficients of the ABCD matrix. We also apply these brackets to the distance-height ray vectors measured from the input and output side of the optical system. We show that these vectors obey a partial Moebius transformation, and that this transformation preserves neither areas nor angles. The Poisson brackets here are expressed in terms of the transverse and longitudinal magnifications.
Arxiv preprint arXiv: …, 2008
We represent vector rotation operators in terms of bras or kets of half-angle exponentials in Cli... more We represent vector rotation operators in terms of bras or kets of half-angle exponentials in Clifford (geometric) algebra Cl 3,0. We show that SO 3 is a rotation group and we define the dihedral group D 4 as its finite subgroup. We use the Euler-Rodrigues formulas to compute the multiplication table of D 4 and derive its group algebra identities. We take the linear combination of rotation operators in D 4 to represent the four Fermion matrices in Sakurai, which in turn we use to decompose any 2 × 2 matrix. We show that bra and ket operators generate left-and right-acting matrices, respectively. We also show that the Pauli spin matrices are not vectors but vector rotation operators, except forσ 2 which requires a subsequent multiplication by the imaginary number i geometrically interpreted as the unit oriented volume.
Arxiv preprint arXiv:0812.0664, 2008
We propose that the height-angle ray vector in matrix optics should be complex, based on a geomet... more We propose that the height-angle ray vector in matrix optics should be complex, based on a geometric algebra analysis. We also propose that the ray's 2 × 2 matrix operators should be right-acting, so that the matrix product succession would go with light's left-to-right propagation. We express the propagation and refraction operators as a sum of a unit matrix and an imaginary matrix proportional to the Fermion creation or annihilation matrix. In this way, we reduce the products of matrix operators into sums of creation-annihilation product combinations. We classify ABCD optical systems into four: telescopic, inverse Fourier transforming, Fourier transforming, and imaging. We show that each of these systems have a corresponding Lagrange theorem expressed in partial derivatives, and that only the telescopic and imaging systems have Lagrange invariants.
Arxiv preprint arXiv:0809.0351, 2008
We list the subgroups of the basis set of Cl 3,0 and classify them according to three criteria fo... more We list the subgroups of the basis set of Cl 3,0 and classify them according to three criteria for construction of universal Clifford algebras: (1) each generator squares to ±1, (2) the generators within the group anticommute, and (3) the order of the resulting group is 2 n+1 , where n is the number of nontrivial generators. Obedient groups we call choirs; disobedient groups, bands. We classify choirs by modes and bands by rhythms, based on canonical equality. Each band generator has a transposition (number of other generators it commutes with). The band's transposition signature is the band's chord. The sum of transpositions divided by twice the number of generator pair combinations is the band's beat. The band's order deviation is the band's disorder. For n ≤ 3, we show that the Cl 3,0 basis set has 21 non-isomorphic subgroups consisting of 9 choirs and 12 bands.
Arxiv preprint arXiv:0810.5224, 2008
We derive the paraxial meridional ray tracing equations from the unified reflection-refraction la... more We derive the paraxial meridional ray tracing equations from the unified reflection-refraction law using geometric algebra. This unified law states that the normal vector to the interface is a rotation of the incident ray or of the refracted ray or of the reflected ray by an angle equal to the angle of incidence or of refraction. We obtain the finite meridional ray tracing equations by simply equating the arguments of the exponential rotation operators. We then derive the paraxial limits of these equations with the help of sign function identities. We show that by embedding the sign functions in the ray tracing equations, we explicitly declare our chosen sign conventions in symbols and not in prose.
Landas: Journal of Loyola School of Theology, 1996
Ionospheric Observation Using FM-CW Radar Array
World Scientific Publishing Co. Pte. Ltd. eBooks, Nov 4, 2010
arXiv (Cornell University), Jul 17, 2008
We derive Copernicus's epicycles from Newton's gravitational force law by assuming that a planet'... more We derive Copernicus's epicycles from Newton's gravitational force law by assuming that a planet's orbit is a perturbed circular orbit, with the perturbation defined to be co-rotating with the said orbit. We substitute this orbit expression into Newton's gravitation law and showed that the perturbation satisfies the linear part of Hill's oscillator equation for lunar motion. We solve this oscillator equation using an exponential Fourier series and impose the boundary conditions at the aphelion and perihelion to derive the Copernicus's formulas for the eccentric, deferent, and epicycle. We show that for small eccetricity, the Copernican orbit expression also leads to Kepler's law of areas for planetary motion. The formalism we use is the Clifford (geometric) algebra Cl 2,0 .
Night-side DP-2 type fluctuations observed by the FM-CW Radar and MAGDAS stations
AGUFM, Dec 1, 2010
DP-2 type fluctuations caused by IMF southward/northward variations have important information ab... more DP-2 type fluctuations caused by IMF southward/northward variations have important information about how the solar wind effects are transferred into the magnetosphere, ionosphere, and on the ground. Since dayside ground magnetic field variations are significantly enhanced during DP-2 events, dayside DP-2 fluctuations have been investigated for many years. However night-side DP-2 variations are not yet investigated sufficiently. In this study,
Philippine Studies: Historical and Ethnographic Viewpoints, Dec 31, 1984
Landas: Journal of Loyola School of Theology, 1992
Apparent Retrograde Motion of Mars Using Geometric Algebra
A planet orbiting the Sun outside the Earth’s orbit moves with a different frequency. When observ... more A planet orbiting the Sun outside the Earth’s orbit moves with a different frequency. When observed at Earth, the movement of the planet does not simply just rise and fall similar to the movement of the Sun and the Moon. Eventually, the superior planet would move “backwards,” or in retrograde, as seen from Earth. In this paper, we will show this apparent retrograde movement using Geometric Algebra.
Estimating abortion rates from contraceptive failure rates via risk compensation: a mathematical model
In this paper, we propose a set of hypotheses for deriving the abortion rate as a function of the... more In this paper, we propose a set of hypotheses for deriving the abortion rate as a function of the inter-course interval in weeks, the number of weeks since the start of first intercourse, the number weeks of pregnancy, the number of weeks of breastfeeding, and the contraceptive failure rate. We also propose risk compensation as feedback: the intercourse interval is proportional to the m th power of the contraceptive failure rate. We show that for different values of m, the abortion rate may become smaller, bigger, or re-main the same compared to the case when no contra-ceptives are used. Thus, one way to settle the RH Bill debate is to determine the correct value of m de-rived from accurate data on the reproductive health parameters of a large sample of the female popula-tion. If this data is not available, it is better not to take risk in approving the bill, because there is a possi-bility of increasing our national abortion rate through the promotion of contraceptives. Instead, it...
A Hestenes Spacetime Algebra Approach to Light Polarization
Applications of Geometric Algebra in Computer Science and Engineering, 2002
Hestenes spacetime algebra is used as a formalism for Light Polarization. A plane-like wave solut... more Hestenes spacetime algebra is used as a formalism for Light Polarization. A plane-like wave solution to the Maxwell equation in free space is studied. Its energy-momentum density is structurally related to the Poincare sphere equation, Stoke’s parameters, and coherency matrices; while its energy momentum is that of light-like particles. The wave is leftcircularly polarized while its spatial inverse is right-circularly polarized. It is conjectured that other polarization states may be obtained by superposing left- and right-ircularly polarized states through Fourier Analysis.
Polarization ellipse and Stokes parameters in geometric algebra
Journal of the Optical Society of America A, 2011
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Papers by Daniel McNamara