Crossing the brachistochrone- Energy of the harmonics n

The energy equation of the brachistochrone we saw in the previous section

does not distinguish between the different harmonics n of the reference photon. But if assume that the wavelength λ of the reference photon in the previous equation refers to some harmonic n, then, if λ₀ is the wavelength of the photon at the first harmonic, n=1, and λ_n is its wavelength at some harmonic n, we have that

where the length L₀≡L_B of the brachistochrone, thus also its radius R₀≡R_B, are constants.

Thus the energy equation, if it expresses the harmonic n, can be written as,

A key element is that the energy E_n, at some harmonic n, will be n² times greater than the energy E₀, at the first harmonic, n=1.

Because of this fact, we may take the following relationships,

Also, if we define the time T_n of the brachistochrone, at some corresponding state n, with respect to the initial time T₀,

then it will be

where

Subsequently, defining the speed v as the ratio

it will also be

If here we set v₀=c, we have the following formulas,

However more generally, from the energy equation we have that

so that the speed of the object v₀ at the first harmonic, n=1, will be equal to the speed of light, only if its mass m₀ is equal to the mass M₀ of the brachistochrone,

Comparing the two speeds, we have

so that it is expected that the speed v_n of the object can exceed of light c, as it moves at higher harmonics.

Such relationships and their consequences will be better understood, after we distinguish between the harmonics and the states of the brachistochrone, in what follows.