Noether's Theorem

Momentum Conservation

For momentum conservation, we can directly start from the EoM: \[ \frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}} = 0 \implies \frac{d}{dt} \frac{\partial L}{\partial \dot{q}} = \frac{\partial L}{\partial q}. \] If \(L\) is not an explicit function of the trajectory \(q\), then \(\frac{\partial L}{\partial q} = 0\), which implies: \[ \frac{d}{dt} \frac{\partial L}{\partial \dot{q}} = 0. \] This means the momentum: \[ p = \frac{\partial L}{\partial \dot{q}} = \text{constant}. \] Another implication of \(\frac{\partial L}{\partial q} = 0\) is that when \(q \to q + \delta q\) (with \(\dot{q}\) unchanged), we have: \[ \delta L = \frac{\partial L}{\partial q} \delta q = 0. \]

Energy Conservation and Hamiltonian

Next, we discuss the Hamiltonian \(H\): \[ \frac{dL}{dt} = \frac{\partial L}{\partial \dot{q}} \frac{d\dot{q}}{dt} + \frac{\partial L}{\partial q} \frac{dq}{dt} + \frac{\partial L}{\partial t}. \] Rearranging and simplifying, we get: \[ \frac{\partial L}{\partial t} = \frac{dL}{dt} - \frac{\partial L}{\partial \dot{q}} \frac{d\dot{q}}{dt} - \frac{\partial L}{\partial q} \dot{q}. \] \[ = \frac{dL}{dt} - \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \dot{q} \right) + \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) \dot{q} - \frac{\partial L}{\partial q} \dot{q}. \] \[ = -\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \dot{q} - L \right) + \left( \frac{d}{dt} \frac{\partial L}{\partial \dot{q}} - \frac{\partial L}{\partial q} \right) \dot{q}. \] The second term corresponds to \(EoM = 0\), so: \[ \frac{\partial L}{\partial t} = -\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \dot{q} - L \right). \] If \(L\) is not an explicit function of time \(t\), then \(\frac{\partial L}{\partial t} = 0\), which implies: \[ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \dot{q} - L \right) = 0. \] The energy: \[ E = \frac{\partial L}{\partial \dot{q}} \dot{q} - L = \text{constant}. \] Another implication of \(\frac{\partial L}{\partial t} = 0\) is that when \(t \to t + \delta t\) (with \(q, \dot{q}\) unchanged), we have: \[ \delta L = \frac{\partial L}{\partial t} \delta t = 0. \]

Global and Local Transformations

In the previous section, we discussed the symmetry of the action \(S\), i.e., \(\delta S = 0\). In this section, we consider \(L\) being independent of \(q\) or \(t\), i.e., \(\delta L = 0\). The difference between the two is reflected in: \[ \delta S = \delta \left( \int L \, dt \right) = \int \delta L \, dt + \int L \delta dt, \] where the second term introduces a difference. From the previous derivation: \[ \int L \delta dt = \int L \frac{d \delta t}{dt} dt. \] If \(\frac{d \delta t}{dt} = 0\), then \(\delta L = 0 \iff \delta S = 0\). A more detailed comparison is shown in the following table:

\[\delta L = 0\]	\[\delta S = 0\]
\[\frac{\partial L}{\partial q} = \frac{d}{dt} \frac{\partial L}{\partial \dot{q}} = 0\]	\[\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \Delta q \right) = 0\]
\[ \frac{\partial L}{\partial t} = -\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \dot{q} - L \right)=0 \]	\[\frac{d}{dt} \left[ \left( \frac{\partial L}{\partial \dot{q}} \dot{q} - L \right) \delta t \right] = 0\]

When the variation \(\Delta q\) is not a function of time (i.e., \(\frac{d}{dt} \Delta q = 0\)), the results of \(\delta L = 0\) and \(\delta S = 0\) are identical. Similarly, when \(\frac{d}{dt} \delta t = 0\), the conservation laws from both approaches are equivalent.

This introduces the concept of global transformations and local transformations. A global transformation assumes the variation is time-independent, meaning the changes at all time points are fixed. A local transformation allows the variation to be time-dependent, providing greater freedom in changes at each time point. Clearly, global transformations are a special case of local transformations. Discussing \(\delta L = 0\) quickly leads to conserved quantities for global transformations, while local transformations pave the way for new directions, such as gauge theory, which we may discuss in future writings.

Originally written in Chinese by the author, these articles are translated into English to invite cross-language resonance.