# Derivation of Noether’s theorem in Lagrangian dynamics

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 q_{i}→q_{i}+εK_{i}(q)

where q denotes the set of all the q_{i}.

*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/∂q_{i})(∂q_{i}/∂ε)+(∂L/∂q_{i}‘)(∂q_{i}‘/∂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/∂q_{i})(dq_{i}/dε)+(∂L/∂q_{i}‘)(dq_{i}‘/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 ∂q_{i}/∂ε=K_{i} and ∂q_{i}‘/∂ε=K_{i}‘.

*This is a bit confusing. Am I correct in thinking that the new q _{i} are the old q_{i}+εK_{i}(q) and here we want ∂[newq_{i}]/∂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 🙂

http://ift.tt/RJL8q6

## Leave a comment