- #1
- 2,959
- 1,505
If one considers the Lagrangian of a non-relativistic particle in a gravitational field,
[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 :)
[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 :)