\(\cfrac { d^{ 2 }x }{ dt^{ 2 } } =\cfrac { 1 }{ m } \psi _{ c } -{ \cfrac { u }{ \psi _{ c }+u } }\cfrac { 1 }{ e^{ u }\left( { e^{ 2u }-1 } \right) ^{ 1/2 } } (\cfrac { dx }{ dt } )^{ 2 }\) ---(*)
where
\(u=ln(cosh(x-\cfrac { x_{ a } }{ 2 } ))\) and
\(\psi_c=\psi _{ n }-\psi_{max}\)
A graph of the math components in the expression is,
We see that all components are positive, given
\(\psi_c=\psi _{ n }-\psi_{max}\gt0\)
\(v_{max}\) is complex when
\(\cfrac { 1 }{ m } \psi _{ c }-\cfrac { d^{ 2 }x }{ dt^{ 2 } } \lt0\)
\(v_{max}=iv\)
this velocity will be in the \(ix\) direction; rotated from\(+x\) by \(\cfrac{\pi}{2}\), anticlockwise.
This can happen when the particle is accelerated further by other factors, a positive temperature particle , a charge, a \(B\) field, etc. The original differential equation will have an additive component that increases acceleration that is independent of \(\psi_n\) but adds energy to the particle,
\(\cfrac { d^{ 2 }x }{ dt^{ 2 } }+A_{other}=\cfrac { 1 }{ m } \psi _{ c } -{ \cfrac { u }{ \psi _{ c }+u } }\cfrac { 1 }{ e^{ u }\left( { e^{ 2u }-1 } \right) ^{ 1/2 } } (\cfrac { dx }{ dt } )^{ 2 }\)
\(\cfrac { d^{ 2 }x }{ dt^{ 2 } }+A_{other}\) is the total acceleration of the particle. If on impact of a photon.
\(\cfrac { 1 }{ m } \psi _{ c }-\cfrac { d^{ 2 }x }{ dt^{ 2 } }-A_{other} \lt0\)
The particle at zero velocity, first goes into a spin, with a velocity component perpendicular to the original direction \(x\), then starts to accelerate along \(x\) as the term \(\cfrac { d^{ 2 }x }{ dt^{ 2 } }\) decreases.
In the case when the particle approaches light speed \(c\) with a very large \(A_{other}\) however, drag due to a space as a very light medium or drag due to entanglement that shares energy with the particle introduces a subtractive component to the original expression,
\(\cfrac { d^{ 2 }x }{ dt^{ 2 } }+A_{other}-D_{other}=\cfrac { 1 }{ m } \psi _{ c } -{ \cfrac { u }{ \psi _{ c }+u } }\cfrac { 1 }{ e^{ u }\left( { e^{ 2u }-1 } \right) ^{ 1/2 } } (\cfrac { dx }{ dt } )^{ 2 }\)
\(D_{other}\) increases with increasing velocity such that at \(v_{max}=c\),
\(A_{other}=D_{other}\)
and
\(\require{cancel}\)
\(\cfrac { 1 }{ m } \psi _{ c }-\cfrac { d^{ 2 }x }{ dt^{ 2 } }-\cancelto{0}{(A_{other}-D_{other}) }\lt0\)
The particle spins at \(c\) then accelerate along \(x\).
Other energy input to the particle receiving a photon, results in spin until the expression,
\(\cfrac { 1 }{ m } \psi _{ c }-\cfrac { d^{ 2 }x }{ dt^{ 2 } }-A_{other} \lt0\)
turns positive with decreasing \(\cfrac { d^{ 2 }x }{ dt^{ 2 } }\).
(*) is a state equation, given \(\cfrac{d^2x}{dt^2}\) and \(\cfrac { 1 }{ m } \psi _{ c }\), if the expression, \(\cfrac { 1 }{ m } \psi _{ c }-\cfrac { d^{ 2 }x }{ dt^{ 2 } }\) is negative, velocity switches to the orthogonal direction.
This is very odd, when velocity and acceleration is perpendicular to each other, as odd as circular motion!
\(\cfrac{d^2x}{dt^2}=\cfrac{v^2}{r}\)
where \(r\) is the radius of the spin from which we may obtain an expression for \(r\),
\(r=\cfrac{1}{\cfrac { \psi _{ c } }{ m v^2} -{ \cfrac { u }{ \psi _{ c }+u } }\cfrac { 1 }{ e^{ u }\left( { e^{ 2u }-1 } \right) ^{ 1/2 } }}\)
Have a nice day.
