How to understand the scope of application of Nott's theorem?

Nott theorem relates symmetry with conserved quantity, which is very useful. It means that every continuous symmetric transformation of a mechanical system has a conserved quantity. Symmetric transformation means that the mechanical system remains unchanged under a certain transformation. Nott theorem is one of the central achievements of theoretical physics, which expresses the one-to-one correspondence between continuous symmetry and conservation law. For example, the laws of physics do not change with time, which means that they have some symmetry about time. If we imagine, for example, that the intensity of gravity changes every day, we will violate the law of conservation of energy, because we can lift heavy objects on the day when gravity is weak and then put them down when gravity is strong, so that we can get more energy than when we started to input. Nott's theorem applies to all physical laws based on the principle of action. It was named after emmy noether, a mathematician in the early 20th century. Nott's theorem is deeply related to quantum mechanics, because it only needs to apply the principles of classical mechanics to identify physical quantities (such as position and momentum) related to Heisenberg's uncertainty principle. Accurately speaking, the word "symmetry" in the above proposition refers to the covariance satisfied by the laws of physics under the action of one-dimensional Lie groups that meet certain technical requirements. The law of conservation of physical quantities is usually expressed by continuity equation. The formal proposition of the theorem only derives the flow expression corresponding to the conserved physical quantity from the invariance condition. This conserved quantity is called Nott charge, and this flow is called Nott flow. Nott flows differ by at most one vector field without divergence. The application of Nott's theorem is helpful for physicists to gain profound insight in any general physical theory by analyzing various transformations that keep the form of relevant laws unchanged. This is called Noether current related to this symmetry. The continuity equation shows that if the flow is integrated on the space slice, a conserved quantity called Nott charge can be obtained (of course, it must be assumed that when m is not tight, the flow falls fast enough when it approaches infinity). Common examples are the relationship between the conservation of momentum, energy and angular momentum and the corresponding space-time uniformity: spatial uniformity and momentum conservation: spatial uniformity, that is, the physical laws on the earth are the same as those on the moon, and the physical laws remain unchanged under the transformation of spatial translation (preferably from the earth to the moon). From Nott's theorem, we can get that there is such a conserved quantity, namely momentum. Spatial isotropy and conservation of angular momentum: space is isotropic, that is, space has no special direction. We take the direction of the coordinate axis at will. Although the values of physical quantities may be different in different coordinate systems, the equations corresponding to physical laws are unchanged. For example, Newton's law of motion F=ma (vector form) remains unchanged under spatial rotation transformation. We rotate the coordinate axis, although the component of the vector has changed, but the overall equation F=ma. Time uniformity and energy conservation are the same: similarly, from time uniformity, that is, the physical laws of the past, present and future are the same, we can get such a conserved quantity-energy from Nott's theorem. The proof of Nott's theorem is generally proved in Lagrangian form, that is, assuming that the Lagrangian description of the mechanical system we found is correct.