I am studying elementary Lagrangian mechanics, and I'm a bit confused about the what's meant by invariance of the Lagrangian under change of coordinates to first order.
More specifically, Noether's theorem states
For each symmetry of the Lagrangian, there is a conserved quantity.
the textbook gives the following definition for symmetry:
By 'symmetry', we mean that if the coordinates are changed by some small quantities, then the Lagrangian has no first-order change in these quantities.
Okay, but it does not mention what is meant by no change in first order. Does it mean all the terms are of first order?
For example, consider the Lagrangian $$L={m \over 2}({\dot x}^2+{\dot y}^2)-{k \over 2}(x^2+y^2)$$ and I let the change of coordinates be $$x \to x+ \epsilon y$$ $$y \to y-\epsilon x$$ and get $${dL \over d\epsilon} = \epsilon k(x^2+y^2)$$
so according to the textbook, a symmetry means no change of Lagrangian in first order of $\epsilon$, thus I should expect ${dL\over d\epsilon} =0$, so does this imply that I should evaluate the derivative at $\epsilon =0$, or is my interpretation problematic?
Best Answer
The appropriate definition of symmetry uses infinitesimal quantities, not just small quantities. Thus, in terms of your question, the Lagrangian is symmetric if $dL/d\epsilon=0$ at $\epsilon=0$.
In terms of your example (rotation of a 2D harmonic oscillator), we have $$ L \to (1+\epsilon^2) L = L + \mathcal{O}(\epsilon^2) $$ Thus to first order in $\epsilon$ $L$ does not change: it's invariant under rotations about the $z$-axis.
So, your interpretation appears correct.