Crossing the brachistochrone- Exponential-sinusoidal driving force

Here an attempt will be made to find an expression for the solution of the driven and damped oscillator

which will be composed of one term (instead of two).

In order to do this we may suppose that the driving force also includes an exponential term,

so that we take a solution in the following compact form

If we substitute this solution into the equation of motion

we will in fact find the same expressions for the amplitude y₀(ω) and phase φ.

We have already done such a calculation. Here we will see how this calculation changes if we suppose that the driving force includes an exponential term, so that we have an equation of motion of the form,

with the following solution,

Substituting this solution into the equation of motion of the driven and damped harmonic oscillator, we take

The exponential term disappears from both sides of the previous equation, so that

This expression is identical to the expression we previously took, without the exponential term.

Thus the exponential- sinusoidal solution

gives the same expressions for the phase φ and the amplitude y₀(ω), with those given by the common steady state solution.

However the exponential- sinusoidal solution has the advantage of ‘remembering’ the initial conditions, because it includes the damping factor in the exponential term. Still it is a solution of the driven damped oscillator, because the amplitude and the phase depend on the driving angular frequency ω.

Letting, for example, the time evolve, t>>1, the exponential term of the solution vanishes, and the solution becomes that of the damped oscillator (without a driving force),

On the other hand, supposing a small damping factor, γ≈0, we take back the standard (steady state) solution of the forced damped oscillator (without the exponential term),

Thus we have a solution which oversees the special cases.

Additionally the assumption of an exponentially- sinusoidal force is not unrealistic, taking into account that the periodic force which is imposed by the moving object onto the oscillating system will not be applied forever, so that it will decrease and vanish in due time.

Here we will consider the exponential- sinusoidal solution, if the frequency ω of the driving force is sufficiently larger than the natural frequency ω₀ of the oscillator.

For simplicity we may set the phase equal to zero, φ=0,

Also the expression for the speed can be simplified as follows,

where we have supposed that here the damping factor γ refers to γ₀.

Considering now the energy corresponding to the exponential- sinusoidal solution for large ω, ω>>ω₀,

we have for the mechanical energy which we may simply call E(t), that,

Thus, if ω>ω₀, the mechanical energy is mostly kinetic energy.

The last expression can also be written as follows,

Some caution is needed with respect to the distinction between the variable damping factor γ(ω), and the constant damping factor γ≡γ₀.

Here is the expanded form of the previous equation,

The energy

can be treated either with respect to the time t, keeping the driving angular frequency ω fixed, or in relation to the frequency ω, keeping the time t fixed, including thus both the aspects of damping, and of a driving force, in a single expression.