Weber-like interactions and energy conservation

TL;DR

This paper investigates Weber-like velocity-dependent forces, demonstrating they are energy-conserving only under specific conditions, and discusses implications for gravitational theories and phenomena like Mercury's perihelion precession and light deflection.

Contribution

It proves the energy conservation condition for Weber-like forces and explores their implications in gravitational theories.

Findings

Weber-like forces are conservative only when =2.

Gravitational theories with Weber-like forces cannot be fully conservative.

Such theories can explain Mercury's perihelion precession and light deflection.

Abstract

Velocity dependent forces varying as $k(\hat{r}/r)(1 - \mu \dot{r}^2 + \gamma r \ddot{r})$ (such as Weber force), here called Weber-like forces, are examined from the point of view of energy conservation and it is proved that they are conservative if and only if $\gamma=2\mu$. As a consequence, it is shown that gravitational theories employing Weber-like forces cannot be conservative and also yield both the precession of the perihelion of Mercury as well as the gravitational deflection of light.