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Particle Mechanics from Local Energy Conservation

Thomas Oikonomou

TL;DR

This work elevates mechanical energy conservation to the foundational principle from which particle dynamics are derived. By enforcing $\dot{E}=0$ with $E=T(v)+V(\mathbf{x})$, it yields a generalized, local force law that couples kinetic energy to inertia without presupposing a specific $T(v)$ or momentum–velocity relation, and it decomposes inertial response into energy-altering and energy-preserving components. Imposing $SO(3)$ symmetry and the Relativity Principle in 1D fixes the velocity response function $f(v)$ under Galilean and Lorentz kinematics, recovering Newtonian and relativistic dynamics as symmetry-realizations of a common energy–force structure. The paper also clarifies the connection to the variational formulation (PLA), identifying when PLA and the energy-conservation (PCE) approach yield equivalent dynamics and highlighting a broader class of energy-conserving dynamics beyond standard PLA. Overall, the framework unifies conservative and non-dissipative single-particle dynamics and delineates precise conditions under which energy-based and variational formulations coincide.

Abstract

We develop a formulation of particle mechanics in which the functional relation between force and kinetic energy is derived directly from local conservation mechanical energy $E$, rather than postulated through Newton's second law or a variational principle. Starting from the instantaneous condition $\dot{E}=0$, imposed as a pointwise constraint along a particle trajectory, we obtain a generalized force law that does not assume a specific kinetic-energy function, momentum-velocity relation, or equation of motion. The resulting inertial response naturally decomposes into a component parallel to the acceleration, responsible for changes in kinetic energy, and a transverse component that preserves energy while altering the direction of motion. Imposing rotational equivariance constrains the geometric structure of the force law, while the relativity principle between inertial reference frames further restricts its admissible realizations. In strictly one-dimensional motion, inertial-frame equivalence implies invariance of the total force under inertial boosts; together with local energy conservation, this uniquely fixes the functional form of kinetic energy and momentum. Galilean invariance selects the Newtonian expressions, whereas Lorentz invariance yields the relativistic ones. The framework unifies conservative and non-conservative (yet non-dissipative) dynamics at the single-particle level and clarifies the precise conditions under which energy-based and variational formulations of mechanics are dynamically equivalent. Newtonian and relativistic mechanics thus emerge as symmetry-selected realizations of a common energy-conserving force-energy structure.

Particle Mechanics from Local Energy Conservation

TL;DR

This work elevates mechanical energy conservation to the foundational principle from which particle dynamics are derived. By enforcing with , it yields a generalized, local force law that couples kinetic energy to inertia without presupposing a specific or momentum–velocity relation, and it decomposes inertial response into energy-altering and energy-preserving components. Imposing symmetry and the Relativity Principle in 1D fixes the velocity response function under Galilean and Lorentz kinematics, recovering Newtonian and relativistic dynamics as symmetry-realizations of a common energy–force structure. The paper also clarifies the connection to the variational formulation (PLA), identifying when PLA and the energy-conservation (PCE) approach yield equivalent dynamics and highlighting a broader class of energy-conserving dynamics beyond standard PLA. Overall, the framework unifies conservative and non-dissipative single-particle dynamics and delineates precise conditions under which energy-based and variational formulations coincide.

Abstract

We develop a formulation of particle mechanics in which the functional relation between force and kinetic energy is derived directly from local conservation mechanical energy , rather than postulated through Newton's second law or a variational principle. Starting from the instantaneous condition , imposed as a pointwise constraint along a particle trajectory, we obtain a generalized force law that does not assume a specific kinetic-energy function, momentum-velocity relation, or equation of motion. The resulting inertial response naturally decomposes into a component parallel to the acceleration, responsible for changes in kinetic energy, and a transverse component that preserves energy while altering the direction of motion. Imposing rotational equivariance constrains the geometric structure of the force law, while the relativity principle between inertial reference frames further restricts its admissible realizations. In strictly one-dimensional motion, inertial-frame equivalence implies invariance of the total force under inertial boosts; together with local energy conservation, this uniquely fixes the functional form of kinetic energy and momentum. Galilean invariance selects the Newtonian expressions, whereas Lorentz invariance yields the relativistic ones. The framework unifies conservative and non-conservative (yet non-dissipative) dynamics at the single-particle level and clarifies the precise conditions under which energy-based and variational formulations of mechanics are dynamically equivalent. Newtonian and relativistic mechanics thus emerge as symmetry-selected realizations of a common energy-conserving force-energy structure.
Paper Structure (10 sections, 63 equations, 2 figures)

This paper contains 10 sections, 63 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Conservation of Energy and the respective Force Law
  3. Relativity Principle and Invariance of the Inertial Response in One Dimension
  4. Determination of $f(v)$ from 1D-FBI under Galilean and Lorentz Kinematics
  5. Galilean Transformations
  6. Lorentz Transformations
  7. Connection between PCE and PLA
  8. Conclusions
  9. Proof of Eq. \ref{['Invar00']}
  10. Proof of Eq. (\ref{['eq41']})

Figures (2)

  • Figure 1: Conceptual Roadmap: Newtonian and relativistic dynamics emerge from local energy balance and 1D Force-Boost Invariance.
  • Figure 2: Roadmap of the structural relations between PCE and PLA. The functions $G$, $P_2$, and $Q_6$ are determined by the scalar response $f(v)$, set by the minimal invariance condition in Eq. \ref{['Invar00']}.