Soumendra
Nath Thakur
Tagore’s Electronic Lab, WB, India
Email: postmasterenator@gmail.com| postmasterenator@telitnetwork.in
Date: June
11, 2025
Quantized
Angular Objects and Time Displacement: Formalization of ∏ᵈᵉᵍ and T(θ°) = Tₓ° in Extended Classical Mechanics
Abstract:
Extended Classical Mechanics
(ECM) requires all mathematical entities to correspond to real, measurable,
physical structures. The abstract constant ∏, commonly regarded as a
dimensionless scalar, is instead formalized here as an angular object ∏ᵈᵉᵍ,
representing the measurable degree-equivalent of one radian. Simultaneously,
the derived relation:
T(θ°) = Tₓ° = θ°/360f = Δt
Interprets angular phase shifts
in real systems as measurable temporal displacements. These formulations extend
ECM's core principle: every mathematical transformation reflects a physical
redistribution—whether of mass, energy, or time—and all quantities must
preserve dimensional identity.
1. Formalization of ∏ᵈᵉᵍ as a Physical Angular Object
In ECM, circular and rotational
motion must reflect real physical angular displacements, not abstract ratios.
Traditionally, ∏ represents the ratio of a circle’s circumference to its
diameter, used across trigonometric and rotational contexts. However, ECM
interprets this ratio in terms of real, countable angular units,
resulting in the definition:
∏ᵈᵉᵍ = 180°/∏ ≈ 57.2948°
This value corresponds to the physical
angular span subtended by one radian in a circle when expressed in degrees.
Rather than treating ∏ as dimensionless, ECM treats ∏ᵈᵉᵍ as an angular object with measurable identity. The
number of such angular units required to span a half-circle becomes:
180°/∏ᵈᵉᵍ ≈ 3.14158
So we express:
180° = 3.14158 × ∏ᵈᵉᵍ
This formulation matches ECM’s
unit consistency protocol and parallels ECM's other physicalised constructs,
such as phase-time shifts and energy-based deformation.
2. Angular Phase Shift as
Temporal Displacement in ECM
Just as angular constants are
converted into physical angular displacements, ECM requires phase shifts to
represent real temporal displacement. When a system oscillates at a frequency f,
and undergoes an angular shift θ° or ₓ°, the
corresponding time shift Δt is given by:
T(θ°) = Tₓ° =
θ°/360f = Δt
Where:
• θ° or ₓ°: Angular
shift in degrees
• f: Oscillatory
frequency (Hz)
• Δt: Actual
physical time delay due to angular offset
This is consistent with earlier
ECM derivations of time modulation due to angular displacement in rotating or
oscillating systems. It physically represents the temporal redistribution
required to generate a phase delay in systems such as waveforms, rotating
fields, or piezoelectric deformations.
Illustrative Example:
For a 90° phase shift at 50 Hz:
T(90°) = 90/(360
× 50) = 1/(4 × 50) = 0.005 sec
This quantifies how an angular
rotation of 90° corresponds to a real delay of 0.005 seconds in the waveform or
rotating field, causing time distortion in the waveform.
3. Physical Implications in ECM Modelling
The dual application of ∏ᵈᵉᵍ and T(θ°) or Tₓ°
supports ECM’s unified treatment of geometry and time:
• In circular or rotational
geometry, ∏ is no longer abstract but counts as ∏ᵈᵉᵍ units of angular displacement.
• In periodic systems, angular
displacements translate into real temporal redistributions, measurable as Δt.
These relations find application
in:
• Rotor and gyroscopic dynamics
• Phase-shifted electrical
signals
• Electromechanical resonance
• Polarized wave front
modulation
• Photon delay or advancement
due to angular phase in ECM field theory
Conclusion:
The reinterpretation of ∏ as ∏ᵈᵉᵍ and the derivation of:
T(θ°) or Tₓ° = θ°/360f or ₓ°/360f
Collectively advance ECM’s central thesis: all observable effects—geometric, temporal, or energetic—must be grounded in real, quantifiable displacements. These constructs replace dimensionless scalars with physically representative units, aligning ECM’s language with observable structure and enforcing continuity across mass, space, and time.
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