How Does the Optico-Mechanical Analogy Explain Light Behavior in Physics?

[Helios]

So, with the mechanical index of refraction

n = $$\sqrt{ 1 - V/E }$$

we plug into the optical ray equation, ( s = arc length )

$$\nabla$$n - [ $$\nabla$$n . ( d$$\vec{r}$$/ds ) ]( d$$\vec{r}$$/ds ) - n ( d$$^{2}$$ $$\vec{r}$$/ ds$$^{2}$$ ) = 0

and get

$$\nabla$$V - [ $$\nabla$$V . ( d$$\vec{r}$$/ds ) ] ( d$$\vec{r}$$/ds ) + 2( E - V )( d$$^{2}$$ $$\vec{r}$$/ ds$$^{2}$$ ) = 0

Now with the replacements

$$\vec{F}$$ = -$$\nabla$$V

( E - V ) = mv$$^{2}$$/2

and get

$$\vec{F}$$ - [ $$\vec{F}$$ . ( d$$\vec{r}$$/ds ) ] ( d$$\vec{r}$$/ds ) - ( mv$$^{2}$$ )( d$$^{2}$$ $$\vec{r}$$/ ds$$^{2}$$ ) = 0

d$$\vec{r}$$/ds = $$\hat{T}$$ is a unit vector tangential to the path

d$$^{2}$$ $$\vec{r}$$/ ds$$^{2}$$ = $$\hat{N}$$/R where $$\hat{N}$$ is a unit normal vector and R is the radius of curvature of the path

So,

$$\vec{F}$$ - ( $$\vec{F}$$ . $$\hat{T}$$ ) $$\hat{T}$$ - ( mv$$^{2}$$/R )$$\hat{N}$$ = 0

mv$$^{2}$$/R is the magnitude of the centripetal force

So with,

$$\vec{F}$$ = F$$_{tangent}$$$$\hat{T}$$ + F$$_{normal}$$$$\hat{N}$$

leads me to believe this derivation is correct. Comments?

 

[Helios]

For the derivation of the ray equation see

https://www.physicsforums.com/showthread.php?t=192736

Also here I present the relativistic version of this formulation. I found this and it might be original.

 

[astrokreng]

The optico-mechanical analogy is a powerful tool used in physics to describe the behavior of light in a mechanical context. In this analogy, light is described as a particle moving along a path, similar to a moving object in mechanics. The equation provided in the content is a manifestation of this analogy, where the index of refraction is linked to the energy and velocity of the particle.

The replacements made to the equation further strengthen this analogy, with the force and centripetal force being described in terms of tangential and normal components, just like in mechanics. This suggests that the derivation is correct and the optico-mechanical analogy can be used to accurately describe the behavior of light.

One potential limitation of this analogy is that it only applies to certain properties of light, such as its path and velocity. It may not fully capture the wave-like nature of light, which is important in other phenomena such as interference and diffraction. Therefore, while the optico-mechanical analogy is a useful tool, it should be used with caution and in conjunction with other theories to fully understand the behavior of light.