# Conservation Law: Invariance Under Translation & Understanding

• vaibhavtewari
In summary: These ICTs correspond to conserved quantities in the system.In summary, if an action is invariant under a translation x->x+a, then there is a conserved current. However, there is no mechanical way to determine if an action possesses a symmetry. One approach is to solve Killing's equations to find all the killing vector fields that generate isometries. In Hamiltonian theory, symmetries can be found by examining the infinitesimal transformations of the dynamical variables in phase space that leave the Hamiltonian unchanged. These correspond to conserved quantities in the system. Overall, there is a powerful theorem that limits the types of symmetries that exist in fundamental theories, but it may still be a difficult task to identify symmetries
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.

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?

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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.

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.

## 1. What is conservation law?

Conservation law is a fundamental principle in physics that states that certain physical quantities, such as energy, momentum, and angular momentum, remain constant over time in a closed system. This means that these quantities cannot be created or destroyed, but can only be transferred or transformed.

## 2. What is meant by invariance under translation?

Invariance under translation is a specific type of symmetry that is found in conservation laws. It means that the laws of physics do not change regardless of where an event takes place in space. This means that the laws are the same no matter where you are in the universe.

## 3. How does the conservation law of invariance under translation affect our understanding of the universe?

The conservation law of invariance under translation is a fundamental principle that guides our understanding of the physical world. It allows us to make predictions about the behavior of objects and systems in different locations in space. It also provides a framework for developing new theories and models to explain natural phenomena.

## 4. Can conservation laws be violated?

While conservation laws are considered fundamental principles, there are some cases where they may appear to be violated. However, these violations can be explained by factors such as measurement errors or the existence of unknown forces. In general, conservation laws have been proven to hold true in a wide range of physical situations.

## 5. How do conservation laws impact our daily lives?

Conservation laws have a significant impact on our daily lives, even if we may not realize it. They are essential for understanding and developing technologies such as energy production, transportation, and communication. They also play a crucial role in fields such as medicine and environmental science, allowing us to make accurate predictions and develop solutions to real-world problems.

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