Conservation Law: Invariance Under Translation & Understanding

[vaibhavtewari]

if action is invariant under a translation x->x+a then there is a conserved current. I was wondering how should we know that if action is invariant for a particular translation in the first place. I know the only way which is when Lagrangian has some canonical variable absent, but this does not look very appealing to me for all the situations. Is there a more encompassing understanding.

kindly share your understanding.

 

[WannabeNewton]

I assume you mean the "lagrangian" that one constructs out of the metric for some space - time since you posted it here? In classical GR, since the "lagrangian" for some particular space- time comes directly from the metric, one systematic way of finding all the conserved quantities is by solving killing's equations $$\bigtriangledown_{\alpha } \zeta _\beta + \bigtriangledown _{\beta}\zeta _{\alpha } = 0$$ therefore finding all the killing vector fields $$\zeta ^{\alpha }$$ which generate all the isometries for that particular space - time such as translations and whatnot. If you mean for any classical lagrangian describing classical systems then you could plug it into the euler - lagrange equations and see if $$\frac{\mathrm{d} }{\mathrm{d} t}(\frac{\partial L}{\partial \dot{x^{\mu }}}) = 0$$ but I guess that is like looking for the cyclic coordinate anyways?

 

[vaibhavtewari]

Thank you for replying, I though understand the stuff you explained, for the benefit of people I will write the reply my professor replied

There is no mechanical way of figuring out if an action possesses a symmetry. Most symmetries in the common actions are easy to recognize, but there are some very subtle ones which are not easy to recognize. Two good things are:
(1) There is a very powerful theorem --- the result of
Haag, Sohnius (sp?) and Lobachevsky (sp?) ---
that there are no coordinate symmetries beyond
supersymmetry; and
(2) The most general local theory with a finite number
of fields, positive energy and diffeomorphism
invariance (or local supersymmetry) is known, as
are all of its symmetries.
But those results apply to fundamental theory. You might always be given an action by some evil physics
professor and asked to find its symmetries, and that might be a tough problem.

 

[Mentz114]

In Hamiltonian theory symmetries can sometimes be found by looking at the infintesimal (contact) transformations (ICT) of the dynamical variables in phase space that leave the Hamiltonian unchanged. If the Hamiltonian has a symmetry group, it can easily be shown that the ICTs have the same commutation relations as the group generators.