Why did Noether use the Lagrangian for her conservation of energy theorem?

Question

So I know that for Noether's conservation of energy theorem, the Lagrangian is used. However, I know that the Lagrangian doesn't always equal energy. So why did she use the Lagrangian and not other representatives of energy (ex. Hamiltonian) to construct her conservation of energy theorem?

Answer 1

However, I know that the Lagrangian doesn't always equal energy.

The Lagrangian never equals the energy, unless there is no potential $U$ so the problem is trivial.

In many cases, we can show that the energy is given by: $$ H = T + U\;, $$ where $T$ is the kinetic energy.

On the other hand, the Lagrangian is given by: $$ L = T - U $$

So why did she use the Lagrangian

Whether or not she did, I don't know. But one would typically start from the Lagrangian equations of motion $$ \frac{d}{dt}\frac{\partial L}{\partial \dot x} = \frac{\partial L}{\partial x}\;, $$ which are a typical starting point for dynamical calculations.

Now consider the total time derivative of the Lagrangian: $$ \frac{dL}{dt} = \frac{\partial L}{\partial x}\dot x + \frac{\partial L}{\partial \dot x}\ddot x + \frac{\partial L}{\partial t}\;.\qquad (1) $$

If the system has no explicit time dependence then: $$ \frac{\partial L}{\partial t}=0 \qquad (2) $$

Combining Eq (1) and (2) tells us that: $$ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot x}\dot x - L\right) = 0\;. $$

Then we define the energy (Hamiltonian) $$ H = \left(\frac{\partial L}{\partial \dot x}\dot x - L\right) $$

So, we have shown that the energy is conserved due to "time translation invariance" of the Lagrangian. It doesn't make sense to do it any other way.

Answer 2

1. The very starting point of Noether's theorem (NT) is an action formulation (and hence a Lagrangian), and quasisymmetries thereof.

2. Noether did not formulate her theorem with only energy conservation in mind. Energy conservation (which according to NT is a consequence of time translation quasisymmetry, cf. e.g. this related Phys.SE post) is just 1 possible conservation law.