Standing Waves, Particles, Time Invariant Fields

Consider the Lagrangian,

$L=T-V$

$\cfrac{d\,}{d\,t}\left(\cfrac{\partial\, L}{\partial\,\dot{x}}\right)=\cfrac{\partial\, L}{\partial\,x}$

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 $x$.