I'm studying the Mirror Symmetry book, available here. I'm reading chapter 10 on quantum mechanics. The following is from page 170. Although this is a basic Lagrangian/Hamiltonian type issue, so the particular book is probably not that important.
We have
\begin{align*} S &= \int dt L \\ &= \int dt (\frac{1}{2} (\frac{d X}{dt})^2 - V(X) ).\tag{10.2} \end{align*}
They say that since the Lagrangian has not explicit $t$ dependence, that the system has time translation symmetry. They say that the action is invariant under $X(t) \to X(t + \alpha)$, for constant $\alpha$.
They then say that if $\alpha$ is now a function of $t$, that
\begin{align*} \delta S &= \int dt \dot{\alpha}(t) (\frac{d X}{dt})^2 + V(X) )\tag{10.5}. \end{align*}
I am having difficulty understanding the argument here, and want to walk through it carefully.
If $\alpha$ is constant, what I think they are saying is that starting from
\begin{align*} S &= \int dt (\frac{1}{2} (\dot{X}(t))^2 - V(X(t)) ),\tag{10.2} \end{align*}
if we consider the same quantity, but with the integrand shifted from $t$ to $t + \alpha$, we can see that
\begin{align*} S &= \int dt (\frac{1}{2} (\dot{X}(t + \alpha))^2 - V(X(t + \alpha)) ) \\ &= \int d(t + \alpha) (\frac{1}{2} (\dot{X}(t + \alpha))^2 - V(X(t + \alpha)) )\\ &= \int dy (\frac{1}{2} (\dot{X}(y))^2 - V(X(y)) )\\ &= \int dt (\frac{1}{2} (\dot{X}(t))^2 - V(X(t)) ). \end{align*}
Right?
Assuming that's what they mean in the first part, now what do they mean in the $\alpha(t)$ case? I think we probably start as above and consider
\begin{align*} \int dt (\frac{1}{2} (\dot{X}(t + \alpha(t)))^2 - V(X(t + \alpha(t))) ). \end{align*}
Then, as above, maybe define a new variable $y = t + \alpha(t)$. So that $dy = dt + \dot{\alpha}(t)dt$.
Then
\begin{align*} \int dt (\frac{1}{2} (\dot{X}(t + \alpha(t)))^2 - V(X(t + \alpha(t))) ) &= \int (dy - \dot{\alpha}(t)dt)(\dot{X}(y))^2 - V(X(y) ) \\ &= \int dy (\frac{1}{2} (\dot{X}(y))^2 - V(X(y)) ) \\ &- \int dt \dot{\alpha}(t)(\frac{1}{2} (\dot{X}(t + \alpha(t)))^2 - V(X(t + \alpha(t))) ) \\ &= \int dt (\frac{1}{2} (\dot{X}(t))^2 - V(X(t)) ) - \int dt \dot{\alpha}(t)(\frac{1}{2} (\dot{X}(t + \alpha(t)))^2 - V(X(t + \alpha(t))) ) \end{align*}
So $$\delta S = - \int dt \dot{\alpha}(t)(\frac{1}{2} (\dot{X}(t + \alpha(t)))^2 - V(X(t + \alpha(t))) .$$ Which is wrong.
Could someone explain what exactly the authors are saying here please?
Also, if possible, if doing a coordinate change, please state the coordinate change as "something = something else". Rather than of the form "something $\to$ something else", because I often find the latter confusing.
Also, I tried to use the author's notation above, but I think, for me at least, there is room for confusion due to the notation $\frac{d}{dt}$ versus "dot". I think in this kind of circumstance where arguments of functions are changing, it might be better to favor notations such as "dot", and take it to mean $\dot{f}$ derivative of the function $f$. For me, $\frac{df}{dt}(g(t))$ is ambiguous, because it could mean derivative with respect to the f, then evaluated at t. Or it could mean derivative wrt t of the composition. So if these sorts of considerations are relevant to answering the above, please spell these things out carefully.
