# 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