Register to reply

Covariance equations of motion and symmetry

by phoenixofflames
Tags: covariance, field theory, symmetry, variation
Share this thread:
phoenixofflames
#1
Jan27-11, 05:38 PM
P: 5
1. The problem statement, all variables and given/known data
Hi, I need to proof the covariance of the equations of motion under an infinitesimal symmetry transformation.


2. Relevant equations
Equations of motion:
[tex]
E_i = \left(\frac{\partial L}{\partial \chi^i}\right) - \partial_{\mu} \left(\frac{\partial L}{\partial \chi^i_{\mu}}\right)
[/tex]
Symmetry transformation
[tex]
\delta \chi^i = \xi^{\alpha} (\chi)
[/tex]
Lagrangian
[tex]
L = L(F^a, \chi^{\alpha}, \chi^{\alpha}_{\mu})
[/tex]

[tex]
\chi^{\alpha}_{\mu} = \partial_{\mu} \chi^{\alpha}
[/tex]



3. The attempt at a solution


[tex]E_i &= \left(\frac{\partial L}{\partial \chi^i}\right) - \partial_{\mu} \left(\frac{\partial L}{\partial \chi^i_{\mu}}\right) [/tex]
[tex]= \left(\frac{\partial L}{\partial \chi^{'\alpha}}\right) \left(\frac{\partial \chi^{'\alpha}}{\partial \chi^i}\right) - \partial_{\mu} \left[\left(\frac{\partial L}{\partial \chi^{' \alpha}_{\beta}}\right) \left(\frac{\partial \chi^{' \alpha}_{\beta}}{\partial \chi^i_{\mu}} \right) \right] [/tex]
[tex]= \left(\frac{\partial L}{\partial \chi^{'i}}\right) + \left(\frac{\partial L}{\partial \chi^{'\alpha}}\right)\left(\frac{\partial \xi^{\alpha}}{\partial \chi^i}\right) - \partial_{\mu} \left[\left(\frac{\partial L}{\partial \chi^{'i}_{\mu}}\right) + \left(\frac{\partial L}{\partial \chi^{' \alpha}_{\mu}}\right) \left(\frac{\partial \xi^{\alpha}}{\partial \chi^i} \right)\right] [/tex]
[tex]= E^{'}_i + \left(\frac{\partial \xi^{\alpha}}{\partial \chi^i} \right) E_{\alpha}[/tex]
at first order in xi.
The answer is
[tex] \delta E_i = - \left(\frac{\partial \xi^{\alpha}}{\partial \chi^i} \right) E_{\alpha}[/tex]
I have no clue actually how to do this...
because L is a function of Chi, but I take the partial derivative towards chi' ,... Actually I have no clue how to do it mathematically correct..
Is it completely wrong or... Is there another way,..
Note that [tex]\delta L[/tex] is not zero and doesn't need to be a complete derivative.

What does this covariance exactly mean?
Phys.Org News Partner Science news on Phys.org
Flapping baby birds give clues to origin of flight
Prions can trigger 'stuck' wine fermentations, researchers find
Socially-assistive robots help kids with autism learn by providing personalized prompts
phoenixofflames
#2
Jan29-11, 03:42 AM
P: 5
Found it by using the action.

Thanks


Register to reply

Related Discussions
What is the difference between dynamical symmetry and geometrical symmetry? Classical Physics 5
Lost in Symmetry and Super Symmetry Quantum Physics 4
Chiral symmetry breaking and approximate flavour symmetry High Energy, Nuclear, Particle Physics 8
Equations of Motion (Deriving equations) Introductory Physics Homework 1
Symmetry Analysis of Partial Differential Equations Differential Equations 5