lagrangian formalism - Prove energy conservation using Noether's theorem

I wonder how you prove that energy is conserved under a time translation using Noether's theorem. I've tried myself but without success. What I've come up with so far is that I start by inducing the following symmetry transformation

$$\begin{align} \mathrm{h}_s:\ &q \rightarrow \mathrm{h}_s(q(t)) = q(t)\\ \hat{\mathrm{h}}_s:\ &\dot{q}(t) \rightarrow \hat{\mathrm{h}}_s(\dot{q}(t)) = \dot{q}(t)\\ &t \rightarrow t^\prime = t+s\epsilon \end{align}$$ $\mathrm{h}_s$ is a symmetry of the Lagrangian if: $$L(\mathrm{h}_s(q(t)),\hat{\mathrm{h}}_s(\dot{q}(t)),t^\prime) = L(x,\dot{x},t) + \frac{\textrm{d}}{\textrm{dt}}F_s$$ Then I derivative with respect to $s$ and look for minimum. $$\frac{\partial}{\partial s}\Big(L(\mathrm{h}_s(q(t)),\hat{\mathrm{h}}_s(\dot{q}(t)),t^\prime) - \frac{\textrm{d}}{\textrm{dt}}F_s\Big)=0$$ I find the derivative to be $$\frac{\partial L}{\partial \mathrm{h}_s(q(t))}\frac{\mathrm{h}_s(q(t))}{\partial s}+\frac{\partial L}{\partial \hat{\mathrm{h}}_s(\dot{q}(t))}\frac{\hat{\mathrm{h}}_s(\dot{q}(t))}{\partial s}+\frac{\partial L}{\partial t^\prime}\frac{\partial t^\prime}{\partial s}- \frac{\textrm{d}}{\textrm{dt}}\frac{\partial F_s}{\partial s}=0$$ $$\Rightarrow \frac{\partial L}{\partial t^\prime}\epsilon-\frac{\textrm{d}}{\textrm{dt}}\frac{\partial F_s}{\partial s} = \frac{\partial L}{\partial t}\frac{\mathrm{dt}}{\mathrm{dt^\prime}}\epsilon -\frac{\textrm{d}}{\textrm{dt}}\frac{\partial F_s}{\partial s} = \frac{\partial L}{\partial t}\epsilon -\frac{\textrm{d}}{\textrm{dt}}\frac{\partial F_s}{\partial s} = 0$$ Here is the part where I get stuck. I don't know what to do next. I'm trying to find my Noether charge that corresponds to a time translation to be the Hamiltonian. Is there an easier or better way to do this? Please teach me, I'm dying to learn!

I found this book, Lanczos, The variational principles of mechanics, page 401, which explicit shows the energy conservation using Noether's theorem. Thou It seems that I can not follow the step from equation 7 to 8. Can someone explain to me why the intregal looks the way it does? Have they taylor expanded the expression somehow?

Answer

Comments to OP's post (v4):

1. 
   

   OP is trying to prove via Noether's theorem that no explicit time dependence of the Lagrangian leads to energy conservation.

   
   


2. 
   

   OP's transformation seems to be a pure horizontal infinitesimal time translation

   $$\tag{A} t^{\prime} - t ~=:~\delta t ~=~-\epsilon, \qquad \text{(horizontal variation)}$$ $$\tag{B} q^{\prime i}(t) - q^i(t)~=:~\delta_0 q^i ~=~0, \qquad \text{(no vertical variation)}$$ $$\tag{C} q^{\prime i}(t^{\prime}) - q^i(t)~=:~\delta q^i ~=~-\epsilon\dot{q}. \qquad \text{(full variation)}$$ It is explained in my Phys.SE answer here why this transformation (A)-(C) cannot be used to prove energy conservation.
   
   
   


3. 
   

   In eq. (1) on p. 401, the Ref. 1 is instead considering the following infinitesimal transformation

   $$\tag{A'} t^{\prime} - t ~=:~\delta t ~=~-\epsilon, \qquad \text{(horizontal variation)}$$ $$\tag{B'} q^{\prime i}(t) - q^i(t)~=:~\delta_0 q^i ~=~\epsilon\dot{q}, \qquad \text{(vertical variation)}$$ $$\tag{C'} q^{\prime i}(t^{\prime}) - q^i(t)~=:~\delta q^i ~=~0. \qquad \text{(full variation)}$$ This is the same infinitesimal transformation as Section IV in my Phys.SE answer here, except for the fact that $\epsilon\equiv\alpha$ is allowed to be a function of time $t$. Therefore the variation of the action $S\equiv A$ is not necessarily zero, but of the form $$\tag{8} \delta S ~=~\int\! dt ~j \frac{d\epsilon}{dt},$$ where the bare Noether current $j=h$ is the energy function, cf. eq. (8) on p. 402 in Ref. 1. The $t$-dependence in $\epsilon$ is tied to the Noether trick explained in this Phys.SE post. This in turn can be pieced together into a proof of the on-shell energy conservation $$\tag{9}\frac{dh}{dt}~\approx~0,$$ cf. eq. (9) on p. 402 in Ref. 1.
   
   

References:

1. C. Lanczos, The variational principles of mechanics, 1970; Appendix II.