If one considers the Lagrangian of a non-relativistic particle in a gravitational field,(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

L = \frac{m}{2}(\delta_{ij}\dot{x}^i \dot{x}^j + 2 \phi(x^k) )

[/tex]

it transforms under

[tex]

\delta x^i = \xi^i (t), \ \ \ \ \delta \phi = \ddot{\xi}^i x_i

[/tex]

as a total derivative:

[tex]

\delta L = \frac{d}{dt}(m \dot{\xi}^i x_i)

[/tex]

My question is: why is the corresponding Noether charge

[tex]

Q = p_i \xi^i - m \dot{\xi}^i x_i

[/tex]

not conserved if one uses the equations of motion? I'm staring at the problem now for quite some time, missing something obvious, but I can't see it :)

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# Conserved Noether charge and gravity

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