Why did Noether use the Lagrangian for her conservation of energy theorem? 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?
 A: 
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.
A: *

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


*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.
