Noether's Theorem to Multi-parameter Transformations

[jstrunk]

When you have single parameter transformations like this in Noether's Theorem
[itex]
\begin{array}{l}
{\rm{ }}t' = t + \varepsilon \tau + ...{\rm{ }}\\
{\rm{ }}{q^\mu }^\prime = {q^\mu } + \varepsilon {\psi ^\mu } + ...
\end{array}
[/itex]

The applicable form of the Rund-Trautman Identity is
[itex]
{\rm{ }}\frac{{\partial L}}{{\partial {q^\mu }}}{\psi ^\mu } + {p_\mu }{{\dot \psi }^\mu } + \frac{{\partial L}}{{\partial t}}\tau - H\dot \tau = \frac{{dF}}{{dt}}
[/itex]

and the conserved quantity is
[itex]
{\rm{ }}{p_\mu }{\psi ^\mu } - H\tau - F.
[/itex]Can someone confirm that with multi-parameter transformations like
[itex]
\begin{array}{l}
{\rm{ }}t' = t + {\varepsilon _i}{\tau _i} + ...{\rm{ }}i = 1,2,...,N\\
{\rm{ }}{q^\mu }^\prime = {q^\mu } + {\varepsilon _i}{\psi _i}^\mu + ...
\end{array}
[/itex]

The Rund-Trautman Identity becomes N identities
[itex]
{\rm{ }}\frac{{\partial L}}{{\partial {q^\mu }}}{\psi _i}^\mu + {p_\mu }{{\dot \psi }_i}^\mu + \frac{{\partial L}}{{\partial t}}{\tau _i} - H{{\dot \tau }_i} = \frac{{d{F_i}}}{{dt}}
[/itex]

and the conserved quantity becomes N conserved quantities
[itex]
{\rm{ }}{p_\mu }{\psi _i}^\mu - H{\tau _i} – F_i.
[/itex]

 

[eljose79]

Yes, your understanding is correct. When there are multiple parameter transformations, the Rund-Trautman Identity becomes N identities and the conserved quantity becomes N conserved quantities. This is because each parameter has its own associated transformation and conserved quantity. Just like in the single parameter case, the conserved quantities can be obtained by solving the N identities.