Editing Angular momentum (section)

=== Stationary-action principle ===
In classical mechanics it can be shown that the rotational invariance of action functionals implies conservation of angular momentum. The action is defined in classical physics as a functional of positions, <math>x_i (t)</math> often represented by the use of square brackets, and the final and initial times. It assumes the following form in cartesian coordinates:<math display="block">S\left([x_{i}];t_{1},t_{2}\right)\equiv\int_{t_{1}}^{t_{2}}d t\left(\frac{1}{2}m\frac{d x_{i}}{d t}\ \frac{d x_{i}}{d t}-V(x_{i})\right)</math>where the repeated indices indicate summation over the index. If the action is invariant of an infinitesimal transformation, it can be mathematically stated as: <math display="inline">\delta S = S\left([x_{i}+\delta x_i];t_{1},t_{2}\right)-S\left([x_{i}];t_{1},t_{2}\right) =0</math>.

Under the transformation, <math>x_i \rightarrow x_i + \delta x_i </math>, the action becomes:
<math display="block">S\left([x_{i}+\delta x_i];t_{1},t_{2}\right)=\!\int_{t_{1}}^{t_{2}}d t\left(\frac{1}{2}m\frac{d(x_{i}+\delta x_{i})}{d t}\frac{d(x_{i}+\delta x_{i})}{d t}-V(x_{i}+\delta x_{i})\right)</math>
where we can employ the expansion of the terms up-to first order in <math display="inline">\delta x_i</math>:
<math display="block">\begin{align}
\frac{d(x_i+\delta x_i)}{d t} \frac{d( x_{i}+\delta x_{i})}{ d t } &\simeq\frac{d x_{i}}{d t} \frac{d x_{i}}{d t}-2\frac{d^{2}x_{i}}{d t^{2}}\delta x_{i}+2\frac{d}{d t}\left(\delta x_{i}\frac{d x_{i}}{d t}\right)\\
V(x_{i}+\delta x_{i}) & \simeq V(x_{i})+\delta x_{i}\frac{\partial V}{\partial x_i}\\
\end{align}</math>giving the following change in action:
<math display="block">S[x_{i}+\delta x_{i}]\simeq S[x_{i}]+\int_{t_{1}}^{t_{2}}d t\,\delta x_{i}\left(- \frac{\partial V}{\partial x_i}-m{\frac{d^{2}x_{i}}{d t^{2}}}\right)+m\int_{t_{1}}^{t_{2}}d t{\frac{d}{d t}}\left(\delta x_{i}{\frac{d x_{i}}{d t}}\right).</math>

Since all rotations can be expressed as [[3D rotation group#Exponential map|matrix exponential]] of skew-symmetric matrices, i.e. as <math>R(\hat n,\theta) = e^{M \theta}</math> where <math>M </math> is a skew-symmetric matrix and <math>\theta </math> is angle of rotation, we can express the change of coordinates due to the rotation <math>R(\hat n,\delta \theta )</math>, up-to first order of infinitesimal angle of rotation, <math>\delta  \theta </math> as:
<math display="block">\delta  x_i  =  M_{ij} x_j \delta \theta . </math>

Combining the equation of motion and '''rotational invariance of action''', we get from the above equations that:<math display="block">0=\delta S=\int_{t_{1}}^{t_{2}}d t\frac{d}{d t}\left(m\frac{d x_{i}}{d t}\delta x_{i}\right)= M_{i j}\,\delta \theta \,  m \,x_{j}\frac{d x_{i}}{d t}\Bigg\vert_{t_{1}}^{t_{2}}</math>Since this is true for any matrix <math>M_{ij} </math> that satisfies <math>M_{ij} = - M_{ji}    , </math> it results in the conservation of the following quantity:
<math display="block">\ell_{ij}(t) := m\left(x_i \frac{dx_j}{dt}-x_j \frac{dx_i}{dt}\right),</math>
as <math>\ell_{ij}(t_1)=\ell_{ij}(t_2)</math>. This corresponds to the conservation of angular momentum throughout the motion.<ref>{{Cite book |last=Ramond |first=Pierre |url=https://books.google.com/books?id=aXr-DwAAQBAJ |title=Field Theory: A Modern Primer |publisher=Routledge |year=2020 |isbn=9780429689017 |edition=2nd}}[https://books.google.com/books?id=aXr-DwAAQBAJ&pg=PA1 Extract of page 1]</ref>