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Obviously like they say hindsight is 20/20, but it seems to me that all the ingredients for Noether's theorem were in place more than a hundred years before its publication, and to be honest it is not a very hard proof. Let me elaborate:

Lagrangian mechanics was formulated in the late 18th century. Just from the Euler-Lagrange equations

$$ \frac{d}{dt} \frac{\partial L}{\partial \dot{q_i}} - \frac{\partial L}{\partial q_i} = 0$$

one can trivially see a basic version of the theorem, which is that if the Lagrangian $L$ is independent of one coordiante $q_i$, the corresponding momentum $p_i = \partial L / \partial \dot{q_i}$ is constant. The more general version is a one-line proof: If the Lagrangian is invariant under a variation $\delta q_i$ of the coordinates and the equations of motion are satisfied, then

$$\delta L = \frac{\partial L}{\partial q_i} \delta q_i + \frac{\partial L}{\partial \dot{q_i}} \frac{d}{dt} \delta q_i = \underbrace{ \left(\frac{\partial L}{\partial q_i} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q_i}}\right) }_{=0} \delta q_i + \frac{d}{dt}\left( \frac{\partial L}{\partial \dot{q_i}} \delta q_i \right) = 0,$$

so that $Q = (\partial L / \partial \dot{q_i}) \delta q_i$ is constant.

Given that this proof uses only theory which was already developed by 1800, and that it is a generalization of a very simple observation, why did it take another hundred years for the theorem to appear?

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  • $\begingroup$ The notion of invariance of differential equations and its relation with the theory of groups and pseudo-groups was only formalized in the late part of the 19th century (e.g. Sophus Lie, Felix Klein, Wilhelm Killing, Elie Cartan, etc.). It seems such a perspective is necessary for formulating Nöther's theorem. $\endgroup$ – Dan Fox Jul 11 '18 at 5:20
  • $\begingroup$ Also, Emmy Noether’s theorems are for lagrangians of arbitrary order with an arbitrary numbers of independent and dependent variables. Your simple observation is for first order lagrangians with one independent variable. $\endgroup$ – Michael Bächtold Jul 17 '18 at 8:05
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I suggest you have a look at the book on Noether Theorems by Yvette Kosmann-Schwarzbach where this history is described in much detail.

There it's clarified how the results you alluded at, were known already to Jacobi (and in the easiest case to Lagrange), so at the beginning of XIXth century.

As Dan mentioned, of course, to formalize such statements in the way Noether did the machinery of groups and group actions was needed. Let me mention the fact that also much work on algebraic invariants (like invariants of quadratic forms) developed between the end of XIXth and the beginning of the XXth were fundamental in inspiring Noether's work.

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