I'm trying to prove via Noether's Theorem that the angular momentum of a massive particle in a gravitational field is conserved. The attempt follows:

OBS: I'm working in the euclidean space so I'll make no distinction between upper and lower indices

The lagrangian of this particle is given by: $L = \frac{1}{2} m \dot{x}^i \dot{x}_i - m \phi(x)$ where $\phi(x)$ is the gravitational potential.

For an infinitesimal rotation by an infinitesimal angle vector $\vec{\theta}$ with components $\theta ^k$ such that $x^i \rightarrow x^i + \epsilon_{ijk}x^j \theta ^k$, we have

\begin{equation} \delta L = - m \partial ^i \phi(x)\epsilon_{ijk}x^j \theta^k + m\dot{x}^i \epsilon_{ijk} \dot{x}^j \theta ^k. \end{equation}

The second term in RHS is zero because $\dot{x}^i \dot{x}^j$ is symmetric and $\epsilon_{ijk}\theta ^k$ is antisymmetric.

My problem comes with the first term in RHS, because I think that it should be zero, such that I would get the conserved quantity $Q = \epsilon_{ijk}m\dot{x}^i x^j \theta^k = \vec{l} \cdot \vec{\theta}$, with $\vec{l}$ being the angular momentum vector, and $\vec{l} \cdot{\vec\theta}$, the angular momentum in the direction of $\theta$. It is necessary that $- m \partial ^i \phi(x)\epsilon_{ijk}x^j \theta^k = 0$ to get conservation of angular momentum or did I make some mistake?

  • $\begingroup$ You are mixing polar and Cartesian coordinates. Work in either one but do not mix. $\endgroup$ – my2cts Mar 15 at 7:32
  • $\begingroup$ Hint: You need to use rotational symmetry of potential. $\endgroup$ – Qmechanic Mar 15 at 7:55
  • $\begingroup$ In $- m \partial ^i \phi(x)\epsilon_{ijk}x^j \theta^k$ we have $\partial ^i \phi(x) = \frac{\partial \phi}{\partial x} \partial ^i x$, where $x = \sqrt{x^a x^a}$, a= 1,2,3, hence $ - m \partial ^i \phi(x)\epsilon_{ijk}x^j \theta^k = - m \frac{\partial \phi}{\partial x} \frac{x^i}{x} \epsilon_{ijk} x^j \theta^k = 0 $, because $\epsilon_{ijk} x^i x^j = 0$. Is this correct? $\endgroup$ – Lil'Gravity Mar 15 at 19:12

You have an extra time derivative in your expression for $Q$, there should be just one. (Your present expression for $Q$ vanishes, for the same reason you have described in the question). Having made that alteration, if we demand that $\frac{dQ}{dt}=0$, we will need to substitute the equation of motion. Upon substitution, this turns out to be precisely the first term.

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