\(\cfrac{d\,T}{d\,t}\)
Given an electron orbital radius, \(r_e\) vs \(T\) curve,
\(\cfrac{d\, r_e}{d\,T}=\cfrac{d\,r_e}{d\,t}.\cfrac{d\,t}{d\,T}\)
A discontinuity in \(\cfrac{d\, r_e}{d\,T}\) will result in a similar break in \(\cfrac{d\,r_e}{d\,t}\).
In this case the time information need to formulate velocity along \(r_e\) is provided for by the rate of temperature change, \(\cfrac{d\,T}{d\,t}\) which is continuous under normal circumstances.
We can also use Hamilton's principle of least action in place of the time information needed.
A electron at an orbital radius \(r_e\) from the nucleus is depicted below,
Is is assume here that the tangential orbital velocity, \(v\) is near terminal speed as a result of drag and does not change with distance \(x\) from the nucleus and time \(t\). It is excluded from the Lagrangian below.
If we formulate the Lagrangian along the radial line,
\(L=T-V\)
\(L=\cfrac{1}{2}m_ev^2_{re}-(-PE_{re})\)
\(PE_{re}\) is negative as zero potential is defined at \(x\rightarrow\infty\); in this system the forces are attractive.
we know that,
\(\cfrac{d}{d\,t}\left\{\cfrac{\partial\,L}{\partial\,\dot x}\right\}=\cfrac{d}{d\,t}\left\{\cfrac{\partial\,}{\partial\,\dot x}\left\{\cfrac{1}{2}m_ev^2_{re}+PE_{re}\right\}\right\}\)
\(=\cfrac{d}{d\,t}\left\{\cfrac{\partial\,}{\partial\,\dot x}\left\{\cfrac{1}{2}m_e(\cfrac { d\, r_{ e } }{ d\, T } \cfrac { d\, T }{ d\, x } \cfrac { d\, x }{ d\, t })^2+PE_{re}\right\}\right\}\)
\(=\cfrac{d}{d\,t}\left\{m_e(\cfrac { d\, r_{ e } }{ d\, T } \cfrac { d\, T }{ d\, x })^2{\dot x}+\cfrac{\partial\,PE_{re}}{\partial\,\dot x}\right\}\)
\(=\cfrac{\partial\,L}{\partial\,x}=\cfrac{\partial}{\partial\,x}\left\{\cfrac{1}{2}m_ev^2_{re}+PE_{re}\right\}\)
\(=\cfrac{\partial}{\partial\,x}\left\{\cfrac{1}{2}m_e(\cfrac { d\, r_{ e } }{ d\, T } \cfrac { d\, T }{ d\, x } \cfrac { d\, x }{ d\, t })^2+PE_{re}\right\}\)
\(=m_{ e }(\cfrac { d\, r_{ e } }{ d\, T } \cfrac { d\, x }{ d\, t } )^{ 2 }\cfrac { d\, T }{ d\, x } \cfrac { d^{ 2 }T }{ d\, x^{ 2 } } +\cfrac { \partial \, PE_{ re } }{ \partial \, x } \)
\(\because \cfrac{d^2\,r_e}{d\,xd\,t}=\cfrac{d\,(\cfrac{d\,r_e}{d\,x})}{d\,t}=0\), \(\cfrac{d^2\,x}{d\,xd\,t}=\cfrac{d\,(\cfrac{d\,x}{d\,x})}{d\,t}=0\)
So,
\(\cfrac { d }{ d\, t } \left\{ m_{ e }(\cfrac { d\, r_{ e } }{ d\, T } \cfrac { d\, T }{ d\, x } )^{ 2 }{ \dot { x } }+\cfrac { \partial \, PE_{ re } }{ \partial \, \dot { x } } \right\} =m_{ e }(\cfrac { d\, r_{ e } }{ d\, T } \cfrac { d\, x }{ d\, t } )^{ 2 }\cfrac { d\, T }{ d\, x } \cfrac { d^{ 2 }T }{ d\, x^{ 2 } } +\cfrac { \partial \, PE_{ re } }{ \partial \, x } \)
We know that \(x\) is a proxy for \(r_e\), ie \(x=r_e\)
\(\cfrac { d }{ d\, t } \left\{ m_{ e }(\cfrac { d\, r_{ e } }{ d\, T } \cfrac { d\, T }{ d\, r_{ e } } )^{ 2 }{ \dot { r_{ e } } }+\cfrac { \partial \, PE_{ re } }{ \partial \, \dot { r_{ e } } } \right\} =m_{ e }(\cfrac { d\, r_{ e } }{ d\, T } \cfrac { d\, r_{ e } }{ d\, t } )^{ 2 }\cfrac { d\, T }{ d\, r_{ e } } \cfrac { d^{ 2 }T }{ d\, r_{ e }^{ 2 } } +\cfrac { \partial \, PE_{ re } }{ \partial \, r_{ e } } \)
since \(\cfrac { \partial \, PE_{ re } }{ \partial \, r_e }=m_e\ddot r_e=F\); this force does work against an opposing force \(-\cfrac { \partial \, PE_{ re } }{ \partial \, r_e }\). The result is an increase in potential energy, as the expression suggests.
\(m_{ e }{ \ddot { r_{ e } } }+\cfrac { d }{ d\, t } \left\{ \cfrac { \partial \, PE_{ re } }{ \partial \, \dot { r_{ e } } } \right\} =m_{ e }(\cfrac { d\, r_{ e } }{ d\, t } )^2\cfrac { d\, r_{ e } }{ d\, T }\cfrac { d^{ 2 }T }{ d\, r^{ 2 }_e } +\cfrac { \partial \, PE_{ re } }{ \partial \,r_e } \)
Using Schwartz Theorem,
\(\cfrac{\partial}{\partial\,\dot r_e}\left\{\cfrac{d\,PE_{re}}{d\,t}\right\}=m_e\dot r^2_{ e }\cfrac { d\, r_{ e } }{ d\, T }\cfrac { d^2 T }{ d\, r^2_e }\)
As when \(\dot r_e=0\), at turning point for, \(r_e\), \(\cfrac{d\,PE_{re}}{d\,t}=0\), we have
\(\cfrac{d\,PE_{re}}{d\,t}=\cfrac{1}{3}m_e\dot r_{ e }^3\cfrac{ d\, r_{ e } }{ d\, T }\cfrac { d^2 T }{ d\, r^2_e }\)
The unit dimension of this expression is consistent. The expression implies that, as the electron moves away from the nucleus, \(PE_{re}\) although always minimum (ie. at \(r_e\), \(PE_{re}\) is a local minima ), is moving from curve to curve with different mimimum value. More importantly, the term \(\cfrac { d\, r_{ e } }{ d\, T }\cfrac { d^2 T }{ d\, r^2_e }\) suggests that there is a discontinuity in \(PE_{re}\) as the electron perform SHM about a mean orbit \(r_e\) over the kink in the profile.
Notice that in the derivation of \(\cfrac{d\,PE_{re}}{d\,t}\) we used the proxy relationship,
\(v_{re}=\cfrac { d\, r_{ e } }{ d\, T } \cfrac { d\, T }{ d\, x } \cfrac { d\, x }{ d\, t }\)
where \(x\) is actually \(r_e\). This method delays the immediate cancellation of certain terms and allows the derivation to evolve until it is convenient to apply the substitution, \(x=r_e\).
