# Why did Noether's theorem take so long to show up?

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?

• 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. – Dan Fox Jul 11 '18 at 5:20
• 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. – Michael Bächtold Jul 17 '18 at 8:05