Let's discuss the physics of a topic I investigated by ''knitting'' together (as I have been accused of before lol) essential parts of questions involving accelerated charges producing Larmor radiation.

A while back, I considered the [contribution] of radiation from the accelerated charges inside of the cavity. This was obtained from a Langrangian that I formed from the ordinary Rayleigh-Plesset equation, but motivated in forming the equation to show strong similarities to the expanding-collapsing model known as the Friedmann equation for cosmology:

[math]mR \ddot{R} + \frac{3}{2}m\dot{R}^2 + \frac{4 \nu m}{R} \dot{R} + \frac{2S m}{\rho_L R} + \frac{\Delta P(t)m}{\rho} = \mathcal{L}[/math]

* Notice, for this to be true, the pressure has units of energy and the denominator is a normal density [math]\frac{\Delta P}{\rho}[/math] with units of velocity squared. We also extract meaning in

[math]\frac{4 \nu m}{R} \dot{R}[/math]

[math]\frac{3}{2}m\dot{R}^2[/math]

In the sense that we know from these two expressions that the viscosity divided by the radius is the same as

[math]\frac{\nu}{R} = \dot{R}[/math]

And notice how similar the form is when dividing through by the radius squared:

[math]m \frac{\ddot{R}}{R} + \frac{3}{2}m(\frac{\dot{R}}{R})^2 + \frac{4 \nu m}{V} \dot{R} + \frac{2S m}{\rho V} + \frac{1}{R^2}\frac{\Delta P(t)m}{\rho} = 0[/math]

This is of course, no coincidence, but both solutions are very closely linked since both solutions pertain to an expanding or collapsing sphere (though many shapes of the universe have been speculated) in fluid dynamics. Notice we can simplify some terms, mainly the density can be extracted on two of the expressions:

[math]m \frac{\ddot{R}}{R} + \frac{3}{2}m(\frac{\dot{R}}{R})^2 + 4 \rho \nu \dot{R} + 2S + \frac{1}{R^2}\frac{\Delta P(t)m}{\rho} = 0[/math]

And so this concludes the first part... even after all this talking, the modified equation still does not describe the radiation given up, the point of this first post was to show the structure of the equation in terms of other things. Next post will describe how charges rotating round a central potential will give up radiation and can be added into the equation as a ''correction term.''

**Edited by Dubbelosix, 08 January 2019 - 03:46 PM.**