\(\psi (x)=-ln(cosh(x-\cfrac { 1 }{ 2 } x_{ a }))+Ax+\psi _{ max }\)
\(\psi_{max}=ln(cosh(\cfrac{1}{2}x_a))\)
From,
\(F=\psi_{n}-\psi(x)=m\cfrac{dv}{dt}=m\cfrac{d^2x}{dt^2}\)
\( \cfrac { d^{ 2 }x }{ dt^{ 2 } } +\cfrac { 1 }{ m } \left\{ -ln(cosh(x-\cfrac { 1 }{ 2 } x_{ a }))+Ax+\psi _{ max } \right\} =\cfrac { 1 }{ m } \psi _{ n }\)
\( \cfrac { d^{ 2 }x }{ dt^{ 2 } } =\cfrac { 1 }{ m } \left\{ \psi _{ n }-\psi _{ max }+ln(cosh(x-\cfrac { 1 }{ 2 } x_{ a }))-Ax \right\} \)
\( \psi _{ c }=\psi _{ n }-\psi _{ max }\)
\( \cfrac { d^{ 2 }x }{ dt^{ 2 } } =\cfrac { 1 }{ m } \left\{ \psi _{ c }+ln(cosh(x-\cfrac { 1 }{ 2 } x_{ a }))-Ax \right\} \)
\( \cfrac { d^{ 2 }x }{ dt^{ 2 } } =\cfrac { 1 }{ 2m } \psi _{ c }t^{ 2 }+\cfrac { 1 }{ m } \iint { \left\{ ln(cosh(x-\cfrac { 1 }{ 2 } x_{ a }))-Ax \right\} } dtdt\)
Let,
\( u=ln(cosh(x-\cfrac { 1 }{ 2 } x_{ a }))-Ax\)
\( \cfrac { d^{ 2 }x }{ dt^{ 2 } } =\cfrac { 1 }{ 2m } \psi _{ c }t^{ 2 }+\cfrac { 1 }{ m } \iint { u } \cfrac { 1 }{ \cfrac { d^{ 2 }x }{ dt^{ 2 } } } \cfrac { d^{ 2 }x }{ dt^{ 2 } } dtdt\)
\( \cfrac { d^{ 2 }x }{ dt^{ 2 } } =\cfrac { 1 }{ 2m } \psi _{ c }t^{ 2 }+\iint { \cfrac { u }{ \psi _{ { c } }+u } } d(dx)\)
\( \cfrac { du }{ dx } =tanh(x-\cfrac { 1 }{ 2 } x_{ a })-A\)
\( \cfrac { dx }{ du } =\cfrac { 1 }{ tanh(x-\cfrac { 1 }{ 2 } x_{ a })-A } \)
\( d(dx)=\cfrac { 1 }{ tanh(x-\cfrac { 1 }{ 2 } x_{ a })-A } dudu\)
\( \cfrac { d^{ 2 }x }{ dt^{ 2 } } =\cfrac { 1 }{ 2m } \psi _{ c }t^{ 2 }+\iint { \cfrac { u }{ \psi _{ { c } }+u } \cfrac { 1 }{ tanh(x-\cfrac { 1 }{ 2 } x_{ a })-A } } dudu\)
\( \cfrac { d^{ 2 }x }{ dt^{ 2 } } =\cfrac { 1 }{ 2m } \psi _{ c }t^{ 2 }+\iint { \cfrac { u }{ \psi _{ { c } }+u } \cfrac { 1 }{ tanh(x-\cfrac { 1 }{ 2 } x_{ a })-A } } \cfrac { du }{ dt } dudt\)
\( \cfrac { dx }{ dt } =\cfrac { 1 }{ m } \psi _{ c }t+\int { \cfrac { u }{ \psi _{ { c } }+u } \cfrac { 1 }{ tanh(x-\cfrac { 1 }{ 2 } x_{ a })-A } } \cfrac { du }{ dt } du\)
\( \cfrac { d(dx) }{ dudt } =\cfrac { 1 }{ m } \psi _{ c }\cfrac { dt }{ du } +{ \cfrac { u }{ \psi _{ { c } }+u } \cfrac { 1 }{ tanh(x-\cfrac { 1 }{ 2 } x_{ a })-A } }\cfrac { du }{ dt } \)
\( \cfrac { d(dx) }{ dudt } \cfrac { du }{ dt } =\cfrac { 1 }{ m } \psi _{ c }\cfrac { dt }{ du } \cfrac { du }{ dt } +{ \cfrac { u }{ \psi _{ c }+u } \cfrac { 1 }{ tanh(x-\cfrac { 1 }{ 2 } x_{ a })-A } }(\cfrac { du }{ dt } )^{ 2 }\)
\( \cfrac { d^{ 2 }x }{ dt^{ 2 } } =\cfrac { 1 }{ m } \psi _{ c }+{ \cfrac { u }{ \psi _{ c }+u } \cfrac { 1 }{ tanh(x-\cfrac { 1 }{ 2 } x_{ a })-A } }(\cfrac { du }{ dx } \cfrac { dx }{ dt } )^{ 2 }\)
Finally,
\( \cfrac { d^{ 2 }x }{ dt^{ 2 } } =\cfrac { 1 }{ m } \psi _{ c }+{ \cfrac { u }{ \psi _{ c }+u } { \left( tanh(x-\cfrac { 1 }{ 2 } x_{ a })-A \right) } }(\cfrac { dx }{ dt } )^{ 2 }\)
where,
\( u=ln(cosh(x-\cfrac { 1 }{ 2 } x_{ a }))-Ax\)
\( \psi _{ c }=\psi _{ n }-\psi _{ max }\)
Compare the above with,
\( \cfrac { d^{ 2 }x }{ dt^{ 2 } } =\cfrac { 1 }{ m } \psi _{ c }-{ \cfrac { u }{ \psi _{ c }+u } }\cfrac { 1 }{ e^{ u }\left( { e^{ 2u }-1 } \right) ^{ 1/2 } } (\cfrac { dx }{ dt } )^{ 2 }\)
Consider,
\( \cfrac { 1 }{ e^{ u }\left( { e^{ 2u }-1 } \right) ^{ 1/2 } } =\cfrac { 1 }{ cosh(x-\cfrac { 1 }{ 2 } x_{ a })sinh(x-\cfrac { 1 }{ 2 } x_{ a }) } ={ 2 }{ csch(2x- x_{ a }) } \)
So,
\( \cfrac { d^{ 2 }x }{ dt^{ 2 } } =\cfrac { 1 }{ m } \psi _{ c }-{ \cfrac { u }{ \psi _{ c }+u } } { 2 }{ csch(2x- x_{ a }) } (\cfrac { dx }{ dt } )^{ 2 }\)
where \(u=ln(cosh(x-\cfrac { 1 }{ 2 } x_{ a }))\)
for the case where the second integration constant was not considered, (ie \(A=0\)). Both the terms, in the respective expressions,
\({ 2 }{ csch(2x- x_{ a }) } \)
\(tanh(x-\cfrac { 1 }{ 2 } x_{ a })-A\)
are positive around \(x_a\).
The conclusions drawn previously for \(A=0\) are applicable for \(0\le A\lt1\).
What would cause \(A\ne0\)? The presence of another particle at a distance \(x\gt2x_z\) or \(x\gt x_a\) where \(x_a\) is the second \(\psi=0\) when \(A=0\).
In the presence of another particle further away, \(\psi\) from the first stretches and envelopes the second particle. The second particle would behave in a reciprocal way, but in our analysis it was considered a point particle.
Interaction of the particles' \(\psi\) is manifested in the constant \(A\).
