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Keywords = Donders’ law

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14 pages, 290 KiB  
Article
Covariant Hamilton–Jacobi Formulation of Electrodynamics via Polysymplectic Reduction and Its Relation to the Canonical Hamilton–Jacobi Theory
by Cecile Barbachoux, Monika E. Pietrzyk, Igor V. Kanatchikov, Valery A. Kholodnyi and Joseph Kouneiher
Mathematics 2025, 13(2), 283; https://doi.org/10.3390/math13020283 - 17 Jan 2025
Cited by 1 | Viewed by 736
Abstract
The covariant Hamilton–Jacobi formulation of electrodynamics is systematically derived from the first-order (Palatini-like) Lagrangian. This derivation utilizes the De Donder–Weyl covariant Hamiltonian formalism with constraints incroporating generalized Dirac brackets of forms and the associated polysymplectic reduction, which ensure manifest covariance and consistency with [...] Read more.
The covariant Hamilton–Jacobi formulation of electrodynamics is systematically derived from the first-order (Palatini-like) Lagrangian. This derivation utilizes the De Donder–Weyl covariant Hamiltonian formalism with constraints incroporating generalized Dirac brackets of forms and the associated polysymplectic reduction, which ensure manifest covariance and consistency with the field dynamics. It is also demonstrated that the canonical Hamilton–Jacobi equation in variational derivatives and the Gauss law constraint are derived from the covariant De Donder–Weyl Hamilton–Jacobi formulation after space + time decomposition. Full article
16 pages, 2361 KiB  
Article
A Non-Inductive Magnetic Eye-Tracker: From Dipole Tracking to Gaze Retrieval
by Valerio Biancalana and Piero Chessa
Instruments 2023, 7(1), 8; https://doi.org/10.3390/instruments7010008 - 7 Feb 2023
Cited by 3 | Viewed by 2571
Abstract
We analyze the information that can be retrieved from the tracking parameters produced by an innovative wearable eye tracker. The latter is based on a permanent-magnet marked corneal lens and by an array of magnetoresistive detectors that measure the magnetostatic field in several [...] Read more.
We analyze the information that can be retrieved from the tracking parameters produced by an innovative wearable eye tracker. The latter is based on a permanent-magnet marked corneal lens and by an array of magnetoresistive detectors that measure the magnetostatic field in several positions in the eye proximity. We demonstrate that, despite missing information due to the axial symmetry of the measured field, physiological constraints or measurement conditions make possible to infer complete eye-pose data. Angular precision and accuracy achieved with the current prototypical device are also assessed and briefly discussed. The results show that the instrumentation considered is suitable as a new, moderately invasive medical diagnostics for the characterization of ocular movements and associated disorders. Full article
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26 pages, 2722 KiB  
Article
Microsaccades, Drifts, Hopf Bundle and Neurogeometry
by Dmitri Alekseevsky
J. Imaging 2022, 8(3), 76; https://doi.org/10.3390/jimaging8030076 - 17 Mar 2022
Cited by 2 | Viewed by 3087
Abstract
The first part of the paper contains a short review of the image processing in early vision is static, when the eyes and the stimulus are stable, and in dynamics, when the eyes participate in fixation eye movements. In the second part, we [...] Read more.
The first part of the paper contains a short review of the image processing in early vision is static, when the eyes and the stimulus are stable, and in dynamics, when the eyes participate in fixation eye movements. In the second part, we give an interpretation of Donders’ and Listing’s law in terms of the Hopf fibration of the 3-sphere over the 2-sphere. In particular, it is shown that the configuration space of the eye ball (when the head is fixed) is the 2-dimensional hemisphere SL+, called Listing hemisphere, and saccades are described as geodesic segments of SL+ with respect to the standard round metric. We study fixation eye movements (drift and microsaccades) in terms of this model and discuss the role of fixation eye movements in vision. A model of fixation eye movements is proposed that gives an explanation of presaccadic shift of receptive fields. Full article
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