# Relationship of conservation laws and symmetries space time

momentum about the axis is conserved. The above three symmetries ( homogeneity and isotropy of space, and homogeneity in time) have never been broken. J. Brau. Physics , Space-Time Symmetries. 4. Conservation Laws. • Continuous symmetries lead to additive conservation laws: – energy, momentum. and observation than in the classical area of space- time symmetry. It may be useful, therefore, to dis- cuss first the relation of phenomena, laws of nature.

It allows investigators to determine the conserved quantities invariants from the observed symmetries of a physical system. Conversely, it allows researchers to consider whole classes of hypothetical Lagrangians with given invariants, to describe a physical system. As an illustration, suppose that a physical theory is proposed which conserves a quantity X. A researcher can calculate the types of Lagrangians that conserve X through a continuous symmetry.

Due to Noether's theorem, the properties of these Lagrangians provide further criteria to understand the implications and judge the fitness of the new theory. There are numerous versions of Noether's theorem, with varying degrees of generality. The original version applied only to ordinary differential equations used for describing distinct particles and not partial differential equations used for describing fields. The original versions also assume that the Lagrangian depends only upon the first derivative, while later versions generalize the theorem to Lagrangians depending on the nth derivative.

Generalizations of Noether's theorem to superspaces also exist. To every differentiable symmetry generated by local actions there corresponds a conserved current.

The word "symmetry" in the above statement refers more precisely to the covariance of the form that a physical law takes with respect to a one-dimensional Lie group of transformations satisfying certain technical criteria.

### Noether's theorem - Wikipedia

The conservation law of a physical quantity is usually expressed as a continuity equation. The first three symmetries lead to three conservation laws: Section IV extends the analysis to symmetry in relativity, showing that these conservation laws exist in that realm. Moreover, for uniform linear motion the symmetry argument applies more elegantly to the Lorentz transformation than to the Galilean transformation. In the following we often talk about variations or changes in the action.

Consistent with standard practice, we will only be interested in variations representing infinitesimal first-order changes in the action.

To keep the arguments simple, we also assume that the particle's invariant mass m rest mass does not change during the motion to be studied. SOFTWARE We start with the well-known definition of action for a particle of mass m that moves from some initial position at time t1 to some final position at time t2: We use the notation KE and PE as symbols for kinetic and potential energies, respectively, because they are more mnemonic than the traditional symbols T and V.

## Noether's theorem

Action is not a familiar quantity 8 for many students, so we employ an interactive computer program 9 to help them develop an intuition about the nature of the action and the principle of least action.

By using an interactive computer display, the student cannot only explore the operation of the principle of least action, but also study the relation between this principle and conservation laws in specific cases Fig.

In carrying out this manipulation, the student naturally works with the central concepts of a worldline a graph of the time dependence of a particle's position and an event a point on a worldline. Unlike the trajectory in space, the worldline specifies completely the motion of a particle.

For background on the symmetry properties of nature, we suggest that our students read a selection from Ref. Translation in space We first examine the symmetry related to translation in space.

When we perform an experiment at some location and then repeat the same experiment with identical equipment at another location, then we expect the results of the two experiments to be the same. So the physical laws should be symmetrical with respect to space translation. As a simple example, consider the action of a free particle in zero potential or uniform potential moving along the x axis between two events 1 [t1,x1] and 2 [t2,x2] infinitesimally close to one another along its worldline.

The principle of least action is symmetrical with respect to a fixed displacement of the position. Noether's theorem implies that this symmetry is connected with a conservation law. In the following, we demonstrate that the conservation law related to symmetry under space translation is conservation of momentum. Principle of least action and momentum Think of the motion of a free particle along the x axis.

To explore the connection between the principle of least action and the conservation of momentum, we take advantage of the additive property of the action to require that the action along an arbitrary infinitesimal section of the true worldline have a minimal value.

Because we are considering translation in space, we fix the first and last events, 1 and 3, and change the space coordinate x2 of the middle event 2 so as to minimize the value of the total action S. This minimum condition corresponds to a zero value of the derivative of S with respect to x2: If we use Eq. The expression on the left-hand side of Eq. We could continue and add other segments C, D, E, to cover the entire worldline that describes the particle motion.

### energy - Spacetime and the conservation laws - Physics Stack Exchange

For all these segments the momentum will have the same value, which yields the conservation law of momentum. The action for this free particle depends only on the change of the coordinate x and the result of this dependence is the conservation of the particle's momentum.

However, this derivation uses only the displacement of one event on the worldline. Therefore, we have not yet demonstrated the relation between the conservation of momentum and the symmetry of translation in space in which all three events are displaced.