An ECM Perspective on Phase Displacement, Source-Dependent Energy Transformation, and Dynamism

 Soumendra Nath Thakur | ORCID: 0000-0003-1871-7803 | Date Oct 30, 2026

Extended Classical Mechanics (ECM) offers a source-originated dynamical interpretation of phase displacement that differs from the transfer-function interpretation considered in this paper - "The Exact Price of a Phase Shift".

In ECM, a phase shift is not merely an abstract phase offset or a mathematical property of a transfer function. It represents a physical-energy difference associated with dynamism and the evolution of a physical source. The accumulated phase displacement is expressed as:

x° = 360° f Δt

where x° represents the accumulated physical-energetic frequency-phase displacement. In the absence of an additional phase displacement, the periodic reference relation is T = 1/f.

ECM identifies two relevant forms of dynamical influence:

1. Motion: The dynamical force relation is Fᴇᴄᴍ = Mᵉᶠᶠaᵉᶠᶠ, with the relevant distance and source-dependent energy difference entering the physical description.
2. Gravitational-potential difference: The corresponding relation is Fɢ = Mᵉᶠᶠgᵉᶠᶠ, with the gravitational-potential difference associated with the relevant source.

In both cases, ECM applies the mass relation:

Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ) = Mɢ

The emergence of Mᵃᵖᵖ corresponds, in the ECM interpretation, to a phase-state potential-energy transformation:

−Mᵃᵖᵖ = ΔPEᴇᴄᴍ = ΔKEᴇᴄᴍ

Here, energy loss refers to a reduction in the potential energy of the relevant source through dynamical transformation, not the destruction of energy or necessarily irreversible dissipation. Total energy remains conserved through the corresponding transformation and transfer.

A physical source is therefore indispensable to this interpretation. For photon emission, the bound electron is the relevant source; for the Planck-frequency threshold, latent frequency is the source. The source is also understood as a physical origin of an existential event, within the broader ECM account of entropic evolution beginning from the very origin.

The associated frequency relationship can be expressed as:

fₛₒᵤᵣ꜀ₑ = fₒᵦₛₑᵣᵥₑᵈ + Δfₛₒᵤᵣ꜀ₑ

Thus, a redshift may be interpreted in ECM as a consequence associated with the source-dependent dynamical energy transformation, rather than as an isolated phase or attenuation parameter.

This distinction is relevant to the scope of the present paper. Its attenuation bound concerns passive, causal, stationary linear time-invariant systems and the mathematical relationship between a specified phase offset and attenuation under the assumptions adopted. It does not incorporate the ECM source-dependent effective-mass relation, the two dynamical force relations, or the associated phase-state energy transformation. Consequently, its result should not be extended without further derivation to moving physical systems or systems subject to gravitational-potential differences.

ECM therefore raises a broader physical question: whether phase displacement, when generated through source-originated dynamism, necessarily involves a source-dependent energy transformation, and how that transformation relates to the attenuation of a particular output channel. This question is distinct from whether a stationary passive transfer function must exhibit attenuation to realise a constant phase offset across a frequency band.