Saved in:
| Main Author: | |
|---|---|
| Format: | Recurso digital |
| Language: | |
| Published: |
Zenodo
2026
|
| Online Access: | https://doi.org/10.5281/zenodo.18881477 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- <p>The shape of the classical Lagrangian \( L = T - V \) is universally used in analytical mechanics and in the variational formulation of physics. Despite its conceptual centrality, this structure is historically introduced as a working hypothesis, not usually being derived from more fundamental principles. In this work, we present a systematic derivation of this form from a minimal interferential structure based on the action phase.</p> <p>We start from three physical hypotheses widely observed in interferential phenomena: (i) the existence of an accumulative quantity associated with trajectories, identified as action \(S\); (ii) the dependence of physical amplitudes on phase factors of the type \(e^{iS/\hbar}\); and (iii) the possibility of parametrizing evolutions by monotonic proper parameters. We initially show that action differences between systems generate isolated episodic coincidences, implying an operational discretization of interferential compatibility events. These events naturally organize into a discrete network of interactions.</p> <p>Under minimal variational hypotheses—global phase invariance, adjacency locality, and flux conservation—we demonstrate that the structural operator associated with the network coincides with the combinatorial Laplacian \( K = D - A \), where \(A\) is the adjacency matrix and \(D\) the degree matrix. The linear dynamics of complex amplitudes on this network, imposing linearity, unitarity, and locality, leads to a unitary generator of the form \( H = \alpha K + \mathrm{diag}(V) \) \cite{vrev}. In the continuous limit, the network Laplacian converges to the differential operator \(-\nabla^2\), recovering the effective Schrödinger equation.</p> <p>The semiclassical decomposition \( \psi = a\,e^{iS/\hbar} \) leads, to leading order in \(\hbar\), to the Hamilton-Jacobi equation \[ \partial_t S + \frac{(\nabla S)^2}{2m} + V = 0, \] from which we identify the effective Hamiltonian \( H = \frac{p^2}{2m} + V \).</p> <p>Applying the Legendre transform, we then obtain the Lagrangian \[ L = \frac{1}{2}m\dot{x}^2 - V, \] showing that the \(T-V\) decomposition emerges as a direct consequence of the phase's interferential structure and the spectral dynamics of the network operator.</p> <p>As a physical test and validation of the formalism, we extend the result to the relativistic regime, identifying the action phase with proper time. We show that the local term of the network generator acts as a modifier of the effective spacetime metric, naturally leading to the isotropic form $g_{00} \approx 1 + 2\Phi/c^2$. Under the stationary phase condition for null geodesics, we derive the exact gravitational refractive index. This produces the Einstein light deflection angle ($4GM/c^2b$) purely as a consequence of phase curvature, without postulating the General Relativity field equations.</p> <p>The central result of the work is that classical mechanics ($L=T-V$) and the geometric structure of gravitation do not need to be assumed as independent postulates, but emerge simultaneously as effective limits of a variational architecture based on phase interference, episodic discretization, and spectral dynamics in networks.</p>