# Lagrangian symmetry problem

## Homework Statement

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.

S = ∫ L dt

## 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...?[/B]

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CAF123
Gold Member
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 $\epsilon$.

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 $\epsilon$.
do you mean taylor expansion??

But then the next part askes me to find conserved quantity. I have no idea how to find it...
A conserved quantity means null time derivative of that quantity. Look at the Euler-Lagrange equation.

CAF123
Gold Member
do you mean taylor expansion??
Yup :)

Yup :)
cool, I will try it first!

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

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

What should be the x?
x is q?

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.