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ZENODO
Other literature type . 2026
License: CC BY
Data sources: ZENODO
ZENODO
Other literature type . 2026
License: CC BY
Data sources: Datacite
ZENODO
Other literature type . 2026
License: CC BY
Data sources: Datacite
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Complex Momentum Theory in Phase Spacetime

Authors: Jarosław Grabiec; Grabiec, Jarosław;

Complex Momentum Theory in Phase Spacetime

Abstract

This work introduces the macroscopic formalism of complex mechanical momentum, in which the active (radial) and reactive (tangential) components are treated as two orthogonal energy channels of orbital motion: active (translational) and reactive (rotational), coupled by a dynamical phase in spacetime. This framework led to the definition of the Triangle of Orbital Impedance (TOI), the Triangle of Orbital Power (TOP), and the Orbital Alignment Factor (OAF). These are the mechanical equivalents of classical structures known from the theory of electric circuits in relation to orbital mechanics. The phase angle, associated with both the impedance and the power distribution, serves as the Orbital Alignment Factor (OAF). This quantitatively describes the instantaneous energy partitioning between radial expansion and rotation and enables the diagnosis of transient states during the expansion of a gravitational system, defining its dynamical stage of development. The Coriolis force takes center stage, interpreted not as an "apparent" component but as a fundamental "player" and internal phase coupling operator. It acts as a lossless mechanical energy transformer between the active and reactive channels while maintaining global consistency with Newton's equations. This formalism leads to the concept of phase orbital spacetime, in which the satellite trajectory is described not only by geometry but also by an internal phase state encoding the energy structure of the system as it evolves in time. The proposed tools (TOI, TOP, OAF) organize energy and power balances in transient states and can provide a better diagnostic tool and a basis for further development of numerical thrust optimization. When extended to gravitational dynamics on larger (cosmological) scales, it can help better describe the dynamic states of celestial mechanics.

Keywords

coriolis force, complex orbital momentum, centrifugal force, momentum, dynamic phase, angular momentum, gravitational tuning coefficient, complex momentum, complex numbers, gravity force, dark energy, orbital reactance, orbit, orbital power factor, complex-valued mechanical momentum, transient state, translational–rotational energy transfer, moment of interia, orbital impedance, Coriolis force coupling, active and reactive mechanical power, non-stationary orbital dynamics, space-time, orbiting, orbital power triangle, rocket, energy channel decomposition (active / reactive), orbital resistance, centripetial force, phase angle of orbital motion, phase-space description of orbits

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
Average
Average
Average