# Noether's theorem problem

1. Feb 7, 2016

### Ziggy12

1. The problem statement, all variables and given/known data
We have the Lagrangian $$L=\frac{1}{2}\dot q^2-\lambda q^n$$
Determine the values for n so that the Lagrangian transform into a total derivative
$$\delta q = \epsilon (t\dot q - \frac{q}{2})$$
2. Relevant equations
The theorem says that if the variation of action
$$\delta S = \int_{t_1}^{t_2} dt\hspace{0.1cm} \delta q \left(\frac{\partial L}{\partial q}- \frac{d}{dt}\frac{\partial L}{\partial \dot q}\right) + \left[\delta q \frac{\partial L}{\partial \dot q}\right ]_{t_1}^{t_2}$$
there is a delta q, and a function lambda such as
$$\delta S = \int_{t_1}^{t_2} dt \frac{d}{dt}\Lambda(q,\dot q)$$
Then $$\epsilon Q(q,\dot q) = \delta q \frac{\partial L}{\partial \dot q} - \Lambda(q,\dot q)$$
is a constant of motion
(At least from what I understood)

3. The attempt at a solution
So I derived the needed derivatives:
$$\frac{\partial L}{\partial q} = -\lambda n q^{n-1}$$
$$\frac{\partial L}{\partial \dot q} = \dot q$$
$$\frac{d}{dt}\frac{\partial L}{\partial \dot q} = \ddot q$$

After some calculations this just gives me formula:
$$\delta S = \epsilon\int_{t_1}^{t_2} \lambda n (\frac{q^n}{2}-t\dot q q^{n-1})+\frac{\dot q^2}{2}+t\ddot q \dot q$$

Getting a function lambda out of this seems impossible, am I doing something wrong here?
Thanks for the help

2. Feb 12, 2016

### Staff: Admin

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

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