\(L=T-V\)
\(\cfrac{d\,}{d\,t}\left(\cfrac{\partial\, L}{\partial\,\dot{x}}\right)=\cfrac{\partial\, L}{\partial\,x}\)
If all dimension are equivalent, then the Lagrangian and Euler-Lagrange Equation is still applicable, when \(t\) is either \(t_c\) or \(t_g\).
\(\psi=T+V=\cfrac{1}{2}m_p\dot{x}^2+V\)
\(L=\cfrac{1}{2}m_p\dot{x}^2-(\psi-\cfrac{1}{2}m_p\dot{x}^2)\)
where \(\psi\) is the total energy between the two space dimensions and \(\dot{x}\) the velocity of the particle.
\(L=m_p\dot{x}^2-\psi\)
We consider existence along \(t_c\),
\(\cfrac{d\,}{d\,t_c}\left(\cfrac{\partial\, L}{\partial\,\dot{x}}\right)=\cfrac{d\,}{d\,t_c}\left(\cfrac{\partial}{\partial\,\dot{x}}\left\{m_p\dot{x}^2-\psi\right\}\right)\)
\(\cfrac { d\, }{ d\, t_c } \left( \cfrac { \partial \, L }{ \partial \, \dot { x } } \right) =\cfrac { d\, }{ d\, t_c } \left( 2m_{ p }\dot { x } -\cfrac { \partial \psi }{ \partial \, \dot { x } } \right) =2m_{ p }\ddot { x } -\cfrac { \partial }{ \partial \, \dot { x } } \left\{ \cfrac { d\, \psi }{ d\, t_c } \right\} \)
and,
\(\cfrac{\partial\, L}{\partial\,x}=\cfrac{\partial}{\partial\,x}\left\{m_p\dot{x}^2-\psi\right\}=-\cfrac{\partial\,\psi}{\partial\,x}\)
So,
\(\cfrac { \partial }{ \partial \, \dot { x } } \left\{ \cfrac { d\, \psi }{ d\, t_c } \right\}=2m_{ p }\ddot { x } +\cfrac{\partial\,\psi}{\partial\,x}\) --- (*)
Differentiate (*) with respect to \(x\),
\(\cfrac { \partial}{ \partial \, x } \left\{ \cfrac { \partial }{ \partial \, \dot { x } } \left\{ \cfrac { d\, \psi }{ d\, t_c } \right\} \right\} =\cfrac { \partial ^{ 2 }\psi }{ \partial \, x\partial \, x } \)
\(\cfrac { \partial }{ \partial \, \dot { x } } \left\{ \cfrac { \partial ^{ 2 }\psi }{ \partial \, t_c\partial \, x } \right\} =\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } \) --- (**)
Differentiate (*) with respect to \(t_c\),
\( \cfrac { \partial }{ \partial \, \dot { x } } \left\{ \cfrac { \partial ^{ 2 }\, \psi }{ \partial \, t^{ 2 }_{ c } } \right\} =2m_{ p }\dddot { x } +\cfrac { \partial ^2 \psi }{ \partial \, x\partial \, t_{ c } } \)
If \(\dddot{x}=0\),
\( \cfrac { \partial }{ \partial \, \dot { x } } \left\{ \cfrac { \partial ^{ 2 }\, \psi }{ \partial \, t^{ 2 }_{ c } } \right\} =\cfrac { \partial ^2\psi }{ \partial \, x\partial \, t_{ c } } \) --- (***)
From the post "Not A Wave But Work Done!", for a wave in \(t_c\), \(t_g\), \(x\), \(x\),
\(\cfrac{\partial^2\psi}{\partial\,t^2_c}=ic\cfrac{\partial^2\psi}{\partial\,x\partial\,t_c}\)
So from (**) and (***),
\( \cfrac { \partial }{ \partial \, \dot { x } } \left\{ ic\cfrac { \partial \, \psi }{ \partial \, x\partial \, t_{ c } } \right\} =ic\cfrac { \partial ^{ 2 }\, \psi }{ \partial \, x^{ 2 } } =\cfrac { \partial }{ \partial \, \dot { x } } \left\{ \cfrac { \partial ^{ 2 }\, \psi }{ \partial \, t^{ 2 }_{ c } } \right\}=\cfrac { \partial ^2 \psi }{ \partial \, x\partial \, t_{ c } } \)
So we have,
\( ic\cfrac { \partial ^{ 2 }\, \psi }{ \partial \, x^{ 2 } } =\cfrac { \partial ^2 \psi }{ \partial \, x\partial \, t_{ c } } \)
\( (ic)^{ 2 }\cfrac { \partial ^{ 2 }\, \psi }{ \partial \, x^{ 2 } } =ic\cfrac { \partial ^2 \psi }{ \partial \, x\partial \, t_{ c } } =\cfrac { \partial ^{ 2 }\, \psi }{ \partial \, t^{ 2 }_{ c } } \)
Is
\(\cfrac { \partial ^{ 2 }\, \psi }{ \partial \, t^{ 2 }_{ c } } =(ic)^{ 2 }\cfrac { \partial ^{ 2 }\, \psi }{ \partial \, x^{ 2 } }=-c^{ 2 }\cfrac { \partial ^{ 2 }\, \psi }{ \partial \, x^{ 2 } }\)
a wave? \(i\) rotates \(x\) into the perpendicular direction. If \(x\) is a radial time than \(ic\) is a circle centered at the origin with \(x\) as radius. We have seen in the post "Gravity Wave and Schumann Resonance", gravity wave of this nature. In the case of an E field, this wave will be perpendicular to the field lines. There is one very important condition in the derivation of this wave, that
\(\dddot{x}=0\),
that all forces effecting \(\psi\),
\(\cfrac{d\,F}{d\,t}=\cfrac{d\,(m\ddot{x})}{d\,t}=m\dddot{x}=0\),
ie. the force field, \(F(x)\) is time invariant. Gravity around a stationary object and the electrostatic force around a stationary charge are example of time invariant field. This condition implies that in \(\psi\),
\(\cfrac{d}{d\,t}\left\{\cfrac{\partial\,\psi}{\partial\,x}\right\}=\cfrac{d\,F(x)}{d\,t}=0\)
The gradient of \(\psi\) does not change with time. BUT
\(ic\cfrac { \partial ^{ 2 }\, \psi }{ \partial \, x^{ 2 } } =\cfrac { \partial ^2 \psi }{ \partial \, x\partial \, t_{ c } }\)
What happen to the wave?! That is not the wave equation in time dimension \(t\). The wave equation in time \(t\) from above is given by,
\(\cfrac { \partial ^{ 2 }\, \psi }{ \partial \, t^{ 2 }_{ c } } =(ic)^{ 2 }\cfrac { \partial ^{ 2 }\, \psi }{ \partial \, x^{ 2 } } =-c^{ 2 }\cfrac { \partial ^{ 2 }\, \psi }{ \partial \, x^{ 2 } } \)
The situation is illustrated in the diagram below,
\(\cfrac{\partial\,\psi}{\partial\,x}\) does not change with time along \(x\).
