From the post "Not A Wave But Work Done!",
\(\ddot { x } \left( 2-i\cfrac { \dot { x } }{ c } \right) \cfrac { \partial \, \psi }{ \partial \, t_{ c } } =ic\left( 1+i\cfrac { \dot { x } }{ c } \right) \dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } +2\ddot { x } \cfrac { \partial V\, }{ \partial \, t_{ c } }\) --- (*)
derived under the assumption that \(\dddot{x}=0\), a time invariant field.
Multiply (*) by \(i\),
\(i\ddot { x } \left( 2-i\cfrac { \dot { x } }{ c } \right) \cfrac { \partial \, \psi }{ \partial \, t_{ c } } =-c\left( 1+i\cfrac { \dot { x } }{ c } \right) \dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } +2i\ddot { x } \cfrac { \partial V\, }{ \partial \, t_{ c } } \)
Multiply by \(\left( 1-i\cfrac { \dot { x } }{ c } \right) \),
\( i\ddot { x } \left( 2-i\cfrac { \dot { x } }{ c } \right) \left( 1-i\cfrac { \dot { x } }{ c } \right) \cfrac { \partial \, \psi }{ \partial \, t_{ c } } =-c\left( 1-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } } \right) \dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } +2i\left( 1-i\cfrac { \dot { x } }{ c } \right) \ddot { x } \cfrac { \partial V\, }{ \partial \, t_{ c } } \)
\( c\left( 1-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } } \right) \dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } =-i\ddot { x } \left( 2-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } } -3i\cfrac { \dot { x } }{ c } \right) \cfrac { \partial \, \psi }{ \partial \, t_{ c } } +2\left( \cfrac { \dot { x } }{ c } +i \right) \ddot { x } \cfrac { \partial V\, }{ \partial \, t_{ c } } \)
\( c\left( 1-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } } \right) \dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } =-\ddot { x } \left\{3\cfrac { \dot { x } }{ c } +i(2-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } } ) \right\} \cfrac { \partial \, \psi }{ \partial \, t_{ c } } +2\left( \cfrac { \dot { x } }{ c } +i \right) \ddot { x } \cfrac { \partial V\, }{ \partial \, t_{ c } } \)
Equating Real terms,
\( c\left( 1-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } } \right) \dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } =-3\cfrac { \dot { x } }{ c } \ddot { x } \cfrac { \partial \, \psi }{ \partial \, t_{ c } } +2\cfrac { \dot { x } }{ c } \ddot { x } \cfrac { \partial V\, }{ \partial \, t_{ c } } \)
\( c\left( 1-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } } \right) \dot { x } \cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } =\cfrac { \ddot { x } }{ c } \left\{-3\cfrac { \partial \, \psi }{ \partial \, t_{ c } } +2\cfrac { \partial V\, }{ \partial \, t_{ c } } \right\} \)
Let's define
\(\cfrac{1}{\gamma^2}=\left( 1-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } } \right) \)
We have,
\( \cfrac{c^2}{\gamma^2} \dot { x } \cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } ={ \ddot { x } } \left\{- 3\cfrac { \partial \, \psi }{ \partial \, t_{ c } } +2\cfrac { \partial V\, }{ \partial \, t_{ c } } \right\} \) --- (**)
Equating Imaginary terms,
\( -\ddot { x } (2-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } } )\cfrac { \partial \, \psi }{ \partial \, t_{ c } } +2\ddot { x } \cfrac { \partial V\, }{ \partial \, t_{ c } } =0\)
\(\ddot{x}\ne 0\)
\( (2-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } } )\cfrac { \partial \, \psi }{ \partial \, t_{ c } }=2\cfrac { \partial V\, }{ \partial \, t_{ c } } \)
\(\cfrac { \partial \, \psi }{ \partial \, t_{ c } } +\left( 1-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } } \right) \cfrac { \partial \, \psi }{ \partial \, t_{ c } }=2\cfrac { \partial V\, }{ \partial \, t_{ c } } \)
\(\left(1+\cfrac{1}{\gamma^2} \right)\cfrac { \partial \, \psi }{ \partial \, t_{ c } }=2\cfrac { \partial V\, }{ \partial \, t_{ c } }\) --- (***)
Substitute the above into (**),
\(\cfrac { c^{ 2 } }{ \gamma ^{ 2 } } \dot { x } \cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } =-{ \ddot { x } }\left\{ 3\cfrac { \partial \, \psi }{ \partial \, t_{ c } } -\left( 1+\cfrac { 1 }{ \gamma ^{ 2 } } \right) \cfrac { \partial \, \psi }{ \partial \, t_{ c } } \right\} \)
\( \cfrac { c^{ 2 } }{ \gamma ^{ 2 } } \dot { x } \cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } =-{ \ddot { x } }\left\{ 2-\cfrac { 1 }{ \gamma ^{ 2 } } \right\} \cfrac { \partial \, \psi }{ \partial \, t_{ c } }\) --- (1)
\({ \ddot { x } }\cfrac { \partial \, \psi }{ \partial \, t_{ c } } =-\cfrac { c^{ 2 } }{ \left\{ 2\gamma ^{ 2 }-1 \right\} } \dot { x } \cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } \)
Equivalently from (1),
\({\left( 1+\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } } \right) \ddot { x } } \cfrac { \partial \, \psi }{ \partial \, t_{ c } } =-c^{ 2 }\left( 1-\cfrac { \dot { x } ^{ 2 } }{ c^{ 2 } } \right) \dot { x } \cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } \)
This suggests that when \(\dot{x}=c\), \(\cfrac{1}{\gamma^2}=0\)
\({ 2\ddot { x } }\cfrac { \partial \, \psi }{ \partial \, t_{ c } }=0\)
Then either,
\(\ddot{x}=0\) and \( \cfrac { \partial \, \psi }{ \partial \, t_{ c } }= \cfrac { \partial \, V }{ \partial \, t_{ c } }\)
or,
\(\cfrac { \partial \, \psi }{ \partial \, t_{ c } }=0\),
that the total energy of the system is a constant in time \(t_c\) and is at an extrema. In this case \(\psi\) has a stable point when \(\dot{x}=c\). From (***),
\(\cfrac { \partial \psi }{ \partial \, t_{ c } }=2\cfrac { \partial V\, }{ \partial \, t_{ c } }=0\)
when \(\dot{x}=c\), and
\(\cfrac { \partial^2 \psi }{ \partial \, t^2_{ c } }=2\cfrac { \partial^2 V\, }{ \partial \, t_{ c }^2 }\)
\(\psi\) is minimum when \(V\) is at its local maximum. This shows that the system, as far as \(\psi\), the total energy is concerned, can be stable at light speed, \(\dot{x}=c\). At that point \(V\) is at its local maximum.
We now consider the case when \(\dot{x}\lt\lt c\),
\({ \ddot { x } } \cfrac { \partial \, \psi }{ \partial \, t_{ c } } =-c^{ 2 } \dot { x } \cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } \)
\({m \ddot { x } } \cfrac { \partial \, \psi }{ \partial \, t_{ c } } =-c^{ 2 }m \dot { x } \cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } \)
With \(F=m\ddot{x}\) and \(p=m\dot{x}\),
\(F\cfrac { \partial \, \psi }{ \partial \, t_{ c } } =-c^{ 2 }p \cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } \)
Since, \(-\cfrac { \partial \, \psi }{ \partial x } =F=m\ddot { x } \)
\(\cfrac { \partial \, \psi }{ \partial x } \cfrac { \partial \, \psi }{ \partial \, t_{ c } } =c^{ 2 }p\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } \)
with \(t_c=\cfrac{1}{\sqrt{2}}t.e^{-i\pi/4}\),
\(\cfrac { \partial \, \psi }{ \partial x } \cfrac { \partial \, \psi }{ \partial \, t } =\cfrac{c^{ 2 }p}{\sqrt{2}}.\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } }.e^{-i\pi/4} \)
in time \(t\).
There should be more implications from this equation to prove its validity or disprove it.
