Proving Symmetry and Finding Conserved Quantities for Lagrangian Functions

[sunmaggot]

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.

Homework Equations

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]

 

[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 $\epsilon$.

 

[sunmaggot]

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??

 

[ddd123]

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]

do you mean taylor expansion??

Yup :)

 

[sunmaggot]

Yup :)

cool, I will try it first!

 

[sunmaggot]

Yup :)

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

 

[ddd123]

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

What should be the x?

 

[sunmaggot]

What should be the x?

x is q?

 

[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.