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Geometrical optics
Published in Timothy R. Groves, Charged Particle Optics Theory, 2017
It is possible to construct a general covariant theory which describes the motion in every reference frame. However, for our purpose here we are interested in the particle motion in a single reference frame which is at rest relative to the laboratory, commonly known as the lab frame. It greatly simplifies the discussion if we confine our attention to this single frame. In the lab frame we can express (2.7) in the equivalent form δ∫tatbLdt=0, where we have defined L = ℒ/γ and t = γτ as the Lagrangian and time, respectively, expressed in the lab frame. The time t is related to the proper time τ by a Lorentz transformation, where we assume the particle coordinate is zero in the particle rest frame. Substituting (2.5) into (2.4), it follows that L(x,v;t)=−mc21−υ2/c2+qv⋅A(x,t)−qϕ(x,t) in the lab frame. We have made use of the vector notation v ∙· A to express the inner product of the two three-vectors v and A. In Cartesian coordinates this is v ∙ A = υxAx + υyAy + υzAz.
Benchmark calculations of the 3 D Rydberg spectrum of beryllium
Published in Molecular Physics, 2022
Monika Stanke, Ludwik Adamowicz
High accuracy atomic calculations also require an accurate account of the leading relativistic and quantum electrodynamics (QED) corrections, as well as the corrections due to the finite mass of the nucleus [6–9]. In our calculations, the account for the effects of the finite nuclear mass is not done is the standard way by employing the perturbation-theory approach, but involves the use in the nonrelativistic variational calculations of a total nonrelativistic Hamiltonian representing the internal state of the system that explicitly depends of the nuclear mass. That Hamiltonian is obtained by separating out the centre-of-mass motion from the laboratory-frame Hamiltonian. The procedure involves transforming the lab-frame Hamiltonian expressed in terms of Cartesian coordinates to a new coordinate system whose first three coordinates are the lab-frame coordinates of the centre of mass and the remaining 3n-3 coordinates are internal coordinates. Some more details about the approach used in our atomic calculations are described in the next section.
Benchmark calculations of the 2 D Rydberg spectrum of lithium
Published in Molecular Physics, 2021
Monika Stanke, Ewa Palikot, Keeper L. Sharkey, Ludwik Adamowicz
The conventional approach to calculate bound states of an atom is to first assume an infinite mass of the nucleus and then correct the results for the nucleus having a finite mass. The perturbation theory method is used to calculate the correction. In our approach, the accounting for the finite nuclear mass starts in the first step of the calculation which is a variational optimisation of the nonrelativistic energy and the corresponding wave function of each considered state of the atom. In this calculation we employ a Hamiltonian that explicitly depends on the masses of all particles forming the atom including the mass of the nucleus. The Hamiltonian is obtained by separating out the operator representing the kinetic energy of the centre-of-mass motion of the atom from its laboratory-frame non-relativistic Hamiltonian. The separation is performed by transforming the Cartesian laboratory coordinate system, in which the lab-frame Hamiltonian is expressed, to a new coordinate system whose first three coordinates are the lab-frame coordinates of the centre-of-mass and the remaining coordinates are the so-called internal coordinates. In the new coordinate system, the total Hamiltonian rigorously separates into the operator representing the kinetic energy of the centre-of-mass motion and the so-called internal Hamiltonian (see the next section). The energies calculated with the finite-nuclear-mass (FNM) approach are specific to a particular isotope of the atom and, thus, allow for a direct calculation of isotope shifts of the total and transition energies.
Comparing inertial measurement units and marker-based biomechanical models during dynamic rotation of the torso
Published in European Journal of Sport Science, 2020
Sara M. Brice, Elissa J. Phillips, Emma L. Millett, Adam Hunter, Bronson Philippa
The raw IMU sensor data were post-processed in Matlab (Mathworks, Natic, USA) using custom scripts. Firstly, the magnetometer data were calibrated using an ellipsoid fitting procedure (Kok & Schön, 2016). A Kalman filter (Kalman, 1960; Maximov, 2018) was used to calculate the orientation of each IMU. The Kalman filter automatically estimates and corrects for any gyroscope bias. The calculated orientations were then transformed from their initial reference frame (which is based upon the gravity vector and the Earth’s magnetic field) to the laboratory reference frame. We calculated the difference between the two reference frames at the start of the trial. Specifically, the lab frame quaternions were calculated as,where superscripts indicate the reference frame, subscripts indicate the source of the measurement, and the parameter t represents time with t = 0 s being the start of the trial. These data were used to compute the relative angle between the torso and pelvis using the same Euler angle rotation sequence used with the marker data. There was four torso mounted IMUs meaning there were four different sets of relative angle data computed.