I'm having a little bit of trouble sorting out the right definitions. I have two sources to work from (but feel free to suggest others I should consult.) . It's in the context of trying to understand Noether's theorem on invariances.
Neuenschwander, Dwight E. "Emmy Noether's wonderful theorem." (2011).
^^^(Note the year: the new edition is much different); and
Arnold, V. I. "Mathematical Methods of Classical Mechanics." Ann Arbor 1001 (1978): 48109.
The context is that of a Lagrangian which is a function of $t$, and dependent variables $q^\mu(t)$ and their derivatives $\dot q^\mu(t)$ (I hope none of the details I am leaving out will get in the way of the question at hand.)
A
Arnold writes:
If $h$ is a homomorphism $M\to M$ on a manifold $M$, we say a Lagrangian system admits $h$ if $L(v) =L(h_\ast v)$ for every $v$ in the tangent space $T(M)$, and $h_\ast:T(M_v)\to T(M_v)$ is the linear map induced by $h$ on $T(M_v)$.
N1
For a sufficiently smooth transformation $\tau$, and $L'=L(\tau(t), \tau(q^\mu), \tau(\dot q^\mu))$, Neuenschwander defines p 64:
We say $\int_a^b L(t, q^\mu,\dot q^\mu)\, dt$ is invariant with respect to $\tau$ if
$\int_a^b L-L'\,dt\sim \epsilon^s$ for $s>1$
N2
Then on page 65:
We say $\int_a^b L(t, q^\mu,\dot q^\mu)\, dt$ is invariant with respect to $\tau$ if
$L'\frac{dt'}{dt}-L\sim \epsilon^s$ for $s>1$
So yes, Neuenschwander offers two distinct, not obviously equivalent definitions of the same term in the span of two pages.
On page 78 he passingly refers to "invariance of the Lagrangian" vs "invariance of the functional" but does not explain himself. In the new edition on page 97 he says a little more, but it's still unclear what is meant. As far as I can see, "invariance of the Lagrangian" was never defined, unless there is a typo in N2.
P
Finally in this solution at physics.se the user refers to the Lagrangian itself being invariant if $L-L'\sim \epsilon^2$ (I don't care much that Neuenschawander uses $s$ and I'm perfectly happy to accept $s=2$ in those definitions.)
I'm trying to reconcile what I see here. Invariance of the Lagrangian itself ($L-L'\sim \epsilon^2$) and invariance of its integral ($\int_a^b L-L'\,dt\sim \epsilon^2)$ are perfectly reasonable ideas to me. I just don't understand which one to pay attention to and how the two ideas are captured in the above definitions.
(The main question): which of these notions am I supposed to pay attention to? Are invariance of the Lagrangian and invariance of its integral of equal importance?
Rewriting the stuff under the integral as $L'\frac{dt'}{dt}-L$ makes sense... but defining invariance of the integral of the Lagrangian with respect to $\epsilon^2$-smallness was not obvious. Perhaps P and N2 amount to the same thing with regards to invariance of the Lagrangian itself?
A does not mention $\epsilon^2$-smallness anywhere... is that perhaps subsumed in the passage from $h$ to the linear map $h_\ast$?
What is the expected relationship between invariance of the Lagrangian vs invariance of its integral? (I thought the first might imply the second, but a friend of mine suggested that this may not be so: his heuristic suggestion was that you could make a lot of tiny changes to the Lagrangian that left it invariant, but which "summed up" by the integral would amount to nontrivial changes. )