# A Noether's Theorem to Multi-parameter Transformations

1. Jul 17, 2016

### jstrunk

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

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

and the conserved quantity is
${\rm{ }}{p_\mu }{\psi ^\mu } - H\tau - F.$

Can someone confirm that with multi-parameter transformations like
$\begin{array}{l} {\rm{ }}t' = t + {\varepsilon _i}{\tau _i} + ...{\rm{ }}i = 1,2,...,N\\ {\rm{ }}{q^\mu }^\prime = {q^\mu } + {\varepsilon _i}{\psi _i}^\mu + ... \end{array}$

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

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

2. Jul 22, 2016