Crossing the brachistochrone- Motion in the brachistochrone

The purpose here is to show that the energy of the brachistochrone is the energy of the (damped and driven) harmonic oscillator.

In the section about the inertial mass- gravitational charge equivalence, we have wondered if in the equation of motion of the driven and damped harmonic oscillator,

the driving force can be equivalent to a gravitational force, of the form

This can be done as follows. First of all in the case of the simple harmonic oscillator we can show that the gravitational force is present, although ‘hidden.’ From the equation of motion of the simple harmonic oscillator we have that

Setting as initial condition (at equilibrium)

the equation of motion transforms into,

so that the energy is

A way to make the factor of ½ disappear is to suppose, for example, that the previous total energy E₀ refers to half a wavelength of the oscillator.

Now we make a step forward to include a damping term in the previous energy equation. The equation of motion of the damped harmonic oscillator is

corresponding to the following total energy

where γ is the damping factor, and ω΄ is the frequency referring to the damping.

For simplicity, we apply the condition of critical damping, so that

where ω΄ is the angular frequency of the damped oscillator, and ω₀ is the angular frequency of the simple (undamped) harmonic oscillator.

In the case of critical damping the total energy takes the following simple form,

In order to include a gravitational term in the previous energy equation, we take the equation of motion of the damped and driven harmonic oscillator,

and apply the initial condition,

where the last term is derived from the simple harmonic oscillator.

Thus the total energy E₀ is given by the following equivalent terms,

Comparing this energy to the energy of the brachistochrone,

we see that it is the same energy, if we identify the amplitude y₀ of the oscillator with the radius R₀ of the brachistochrone, so that,

Also, if we have critical damping, then v₀=c, so that the speed of the object moving on the brachistochrone at the first state (n=1) can be substituted by the speed of light,

The point is that since the energy of the brachistochrone is the same with the energy of the damped and forced harmonic oscillator, then the equations of motion which apply to the oscillator will also apply to the brachistochrone.

Notes:

With respect to the harmonic (elastic) term kR₀², and the damping term m(γ²/4)R₀², which appear in the energy equation of the brachistochrone,

we can define the harmonic constant k, and the damping factor γ, in such a way that, from the energy equation of the brachistochrone, at any state n,

it will be

where also

since

Besides the aspect that the constant k_n here depends on the state of the system n, its particularity is that it relates the microscopic aspects (the wavelength λ of the reference photon) to the macroscopic aspects (the radius R of the brachistochrone) of the system. Therefore it is something more than a ‘spring constant.’

Additionally, an approach to the damping factor γ can be made, if from the energy equation of the brachistochrone,

we use the terms

so that, it will be

Having the condition of critical damping, γ₀=2ω₀, v₀=c, we also have that

Thus, for the damping factor γ_n we take

where

Thus we can define the following products,

Introducing a time t which comes in multiples n of the initial period T₀,

we take

where the time t was defined in such a way that the radian factor 4π disappears from the previous products.

This result is revealing in the sense that the previous products give us either n, or n², depending on whether we use the damping factor γ at the ground state (n=1), or at some higher state n, respectively.

The previous products and the introduction of the time t will be helpful in order to the express the energy of the brachistochrone in an exponential form, later on.