Derivation of Noether’s theorem in Lagrangian dynamics

I’m going to run through a derivation I’ve seen and ask a few questions about some parts that I’m unsure about.

Firstly the theorem: For every symmetry of the Lagrangian there is a conserved quantity.

Assume we have a Lagrangian L invariant under the coordinate transformation qi→qi+εKi(q)
where q denotes the set of all the qi.
My book says that by a symmetry we mean that L is unchanged to first order when the coordinates are changed by some small amount. This doesn’t really fit in with my idea of symmetry. I don’t really understand the physical idea of the statement.

This means that dL/dε=0, but dL/dε=∑i[(∂L/∂qi)(∂qi/∂ε)+(∂L/∂qi‘)(∂qi‘/∂ε)]. I’m using primes instead of dots for time derivatives by the way.
My first issue is that we’re finding the total derivative dL/dε, so why does the chain rule not take the form ∑i[(∂L/∂qi)(dqi/dε)+(∂L/∂qi‘)(dqi‘/dε)]?
Secondly can’t L have time dependence, so what happens to the term (∂L/∂t)(∂t/∂ε)?
Is it because the term is zero because obviously changing the coordinates by some amount ε has no influence on the passage of time?

Now substitute ∂qi/∂ε=Ki and ∂qi‘/∂ε=Ki‘.
This is a bit confusing. Am I correct in thinking that the new qi are the old qi+εKi(q) and here we want ∂[newqi]/∂ε etc which gives the result stated above when using the relation stated. Obviously if these were total derivatives things would be messy.

Then we just use the EL equation on the first term, and note we have what looks like the derivative of a product. We can write this as d/dt(conserved quantity)=0 to prove the theorem.

Thankyou to anybody who helps 🙂

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