Symmetry of a lagrangian & Noether's theorem

[irycio]

Homework Statement

Assuming that transformation q->f(q,t) is a symmetry of a lagrangian show that the quantity
$$f\frac{\partial L}{\partial q'}$$ is a constant of motion ($$q'=\frac{dq}{dt}$$).

2. Noether's theorem
http://en.wikipedia.org/wiki/Noether's_theorem

The Attempt at a Solution

Now, what I guess is that this exercise isn't well-formulated. Noether's theorem is obviously first thing that comes to our minds. Quantity mentioned above, however, would be a constant of motion, if the transformation was q->q+f(q,t), not simply q->f(q,t), since the latter transformation leads us to q->q+(f(q,t)-q), and the conserved quantity $$(f-q)\frac{\partial L}{\partial q'}$$.

Am I right and there is a tiny mistake in the exercise or there is a way to show that the given quantity is really conserved? (I tried and failed).

 

[sgd37]

Yeah I think they did mean q-> q+f(q,t), and it smells a lot like q->exp(iat)q

 

[sgd37]

Otherwise since q+dq=f(q,t) then dq = t*df/dt