Editing Noether's theorem (section)

=== Geometric derivation ===

The Noether’s theorem can be seen as a consequence of the [[fundamental theorem of Calculus#Generalizations|fundamental theorem of calculus]] (known by various names in physics such as the [[Generalized Stokes theorem]] or the [[Gradient theorem]]):<ref>{{cite journal | last= Houchmandzadeh |first=B. | year = 2025 | title = A geometric derivation of Noether's theorem | journal = European Journal of Physics | volume = 46 | pages = 025003 |url= https://hal.science/hal-04682603v3/document}}
</ref> 
for a function 
<math display="inline">S</math> 
analytical in a domain 
<math display="inline">{\cal {D}}</math>,
<math display="block">\int_{{\cal {\cal P}}}dS=0</math> 
[[File:Geometric derivation of Noether's theorem.svg|thumb|Integration path that leads to Noether's theorem]]
where <math display="inline">{\cal P}</math> is a closed path in <math display="inline">{\cal D}</math>. Here, the ''function'' <math display="inline">S(\mathbf{q},t)</math> is the action ''function'' that is computed by the integration of the Lagrangian over optimal trajectories or equivalently obtained through the [[Hamilton-Jacobi equation]]. As <math display="inline">\partial S/\partial\mathbf{q}=\mathbf{p}</math> (where <math display="inline">\mathbf{p}</math>is the momentum) and <math display="inline">\partial S/\partial t=-H</math> (where <math display="inline">H</math> is the Hamiltonian), the differential of this function is given by <math display="inline">dS=\mathbf{p}d\mathbf{q}-Hdt</math>.

Using the geometrical approach, the conserved quantity for a symmetry in Noether’s sense can be derived. The symmetry is expressed as an infinitesimal transformation:<math display="block">\begin{aligned}
\mathbf{q'} & = & \mathbf{q}+\epsilon\phi_{\mathbf{q}}(\mathbf{q},t)\\
t' & = & t+\epsilon\phi_{t}(\mathbf{q},t)
\end{aligned}</math>  Let <math display="inline">{\cal C}</math> be an optimal trajectory and <math display="inline">{\cal C}'</math> its image under the above transformation <math display="inline">(\phi_{\mathbf{q}},\phi_{t})^{T}</math> (which is also an optimal trajectory). The closed path <math display="inline">{\cal P}</math> of integration is chosen as <math display="inline">ABB'A'</math>, where the branches <math display="inline">AB</math> and <math display="inline">A'B'</math> are given <math display="inline">{\cal C}</math> and <math display="inline">{\cal C}'</math> . By the hypothesis of Noether theorem, to the first order in <math display="inline">\epsilon</math>, <math display="block">\int_{{\cal C}}dS=\int_{{\cal C}'}dS</math> therefore, <math display="block">\int_{A}^{A'}dS=\int_{B}^{B'}dS</math> By definition, on the <math display="inline">AA'</math> branch we have <math display="inline">d\mathbf{q}=\epsilon\phi_{\mathbf{q}}(\mathbf{q},t)</math> and <math display="inline">dt=\epsilon\phi_{t}(\mathbf{q},t)</math>. Therefore, to the first order in <math display="inline">\epsilon</math>, the quantity <math display="block">I=\mathbf{p}\phi_{\mathbf{q}}-H\phi_{t}</math> is conserved along the trajectory.