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Lagrangian symmetry problem

  • Thread starter sunmaggot
  • Start date
  • #1
62
4

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]
 
Last edited:

Answers and Replies

  • #2
CAF123
Gold Member
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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
62
4
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??
 
  • #4
481
55
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.
 
  • #5
CAF123
Gold Member
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88
do you mean taylor expansion??
Yup :)
 
  • #6
62
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  • #7
62
4
Yup :)
the taylor series has a term (x-a), what should be the a?
 
  • #8
481
55
the taylor series has a term (x-a), what should be the a?
What should be the x?
 
  • #9
62
4
  • #10
481
55
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.
 

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