# Lagrangian symmetry problem

1. Sep 28, 2015

### sunmaggot

1. The problem statement, all variables and given/known data
Given L (q, dq/dt, t).
translation: q ---> q + e (e is infinitesimal constant)
show that if ∂L/∂q = 0, then L is symmetry under the above translation.
then find conserved quantity.

2. Relevant equations
S = ∫ L dt

3. The attempt at a solution
My attempt is nothing... because I don't know the proper procedure to prove a symmetry. Do I simply prove L' - L = 0? But then the next part askes me to find conserved quantity. I have no idea how to find it...
So, can anyone spare me some hints...?

Last edited: Sep 28, 2015
2. Sep 28, 2015

### CAF123

Let $L_1=L_1(q, \dot{q}, t)$ and $L_2$ that with the replacement $q\rightarrow q+ \epsilon$. Now consider the expansion of $L_2$ in the infinitesimal parameter $\epsilon$.

3. Sep 28, 2015

### sunmaggot

do you mean taylor expansion??

4. Sep 28, 2015

### ddd123

A conserved quantity means null time derivative of that quantity. Look at the Euler-Lagrange equation.

5. Sep 28, 2015

### CAF123

Yup :)

6. Sep 28, 2015

### sunmaggot

cool, I will try it first!

7. Sep 28, 2015

### sunmaggot

the taylor series has a term (x-a), what should be the a?

8. Sep 28, 2015

### ddd123

What should be the x?

9. Sep 28, 2015

### sunmaggot

x is q?

10. Sep 28, 2015

### ddd123

That's not what CAF123 suggested you to do. L is a function of phase space: you are interested in a small perturbation of the position variable, which changes the whole function L. So this means taking a McLaurin expansion in the infinitesimal translation parameter.