Noether's theorem

The Noether theorem (formulated in 1918 by Emmy Noether ) linked elementary physical concepts such as charge, energy and momentum with geometric properties, the invariance of the action under symmetry transformations:

It is a symmetry transformation (for example a rotation or displacement ) that does not change the behavior of the physical system.

A conserved quantity of a system of particles is a function of time, place of the particles and their speed, whose value does not change on each physically traversed path with time. For example, the energy of a particle of mass moving in a potential, a conserved quantity, that is true for all time

Examples of symmetries and conserved quantities associated

  • From the homogeneity of the time (selects the start time does not matter) the conservation of energy ( conservation of energy ) follows. Thus, the energy of the pendulum when neglecting friction remains the same, but not the energy of a swing, to the changed a child by lifting and lowering his body, the length of the suspension to the center of gravity in time.
  • From the homogeneity of space ( choice of starting location does not matter) results in the conservation of momentum ( conservation of momentum ). Thus, the pulse of a free particle is constant, but not the momentum of a particle in the gravitational field of the Sun; its location is important for the movement of the particle. Because a free particle of mass unchanged moves with uniform velocity, when looking at a uniformly moving observers, the weighted starting point, is a conserved quantity. Generalized to multiple particles follows that the center of gravity moves with uniform velocity when the total force vanishes.
  • From the isotropy of space, namely the rotational invariance ( direction in space does not matter), results in the conservation of angular momentum ( conservation of angular momentum ). Thus, the angular momentum of a particle in the gravitational field of the sun is retained, because the gravitational potential is the same in all directions.

The symmetries that belong to the conservation of electric charge and other charges of elementary particles, affect the wave functions of electrons, quarks and neutrinos. Each said charge is a lorentzinvarianter scalar, that is, it has the same value in all reference frames, unlike, for example, angular momentum, or the energy of the pulse.

Mathematical formulation


The formulated in Noether 's theorem relating symmetries and conservation laws apply to such physical systems whose motion or field equations can be derived from a variational principle. The differential equations to be met by such systems, indicate that the variation of an action functional vanishes.

With the movement of mass points this effect by a Lagrangian functional is the time, place and speed

Characterized and classified each differentiable trajectory, the time integral

About. For example, in Newtonian physics, the Lagrangian of a particle in a potential difference of kinetic and potential energy

The physico actually traversed path that goes to the beginning of time through the start point and the end time by the target, makes the value of the effect in comparison with all other ( differentiable ) paths that go well initially, and finally through, stationary. Therefore actually physically traversed path satisfies the equation of motion ( see derivation of Lagrange formalism )

For motion of a particle in the potential, this is the Newtonian equation of motion

Differential equations, which can be derived by varying such a functional effect is called variationally selfadjoint. All elementary field and motion equations of physics are self-adjoint variational.


It is said that the differential equation has a symmetry, if there is a transformation of the space of the curves depicting the solutions of the differential equations solutions. For variationally selfadjoint differential equations one obtains such a transformation if the transformation the action functional invariant can be up to boundary terms. The Noether theorem states that the invariance of the action functional with respect to a one-parameter continuous group of transformations, the existence of a conserved quantity for the order and that, conversely, every conserved quantity, the existence of ( at least infinitesimal ) symmetry of the action has a consequence.

We restrict ourselves to symmetries in classical mechanics. For the field-theoretical proof we refer to the English version of the article.

Be a one-parameter differentiable group of transformations, the ( sufficiently differentiable ) mapping curves on curves and belong to the parameter value to the identity map.

For example, forms with each curve from the order previously traversed curve. Transforming each curve to shift to a constant.

The transformations are called local if the derivative at the identity map, the infinitesimal transformation

For all curves can be as a function of time, place and speed, evaluated on the curve, writing,

For example, the shifts of time and place locally and belong to the infinitesimal transformation or to.

Let now the Lagrangian of the mechanical system. Then the names of the local transformations symmetries of the action, if the Lagrangian at infinitesimal transformations only to the time derivative of a function evaluated for all curves, changes,

Because then, the action changes only by boundary terms

The context of this definition, the symmetry of the interaction with the conserved quantity is clear when one performs the partial derivatives of the Lagrangian after, and thereby used as a shorthand definition of the infinitesimal transformation

Adds the first term to a multiple of the equation of motion and moves to the addition of the second term from one arises

And the defining equation of an infinitesimal symmetry of an effect is

Since, however, disappears, the times of the equations of motion on the physical paths traversed, this equation states that the function of

Belonging to the symmetry Noether charge, does not change on the physically traversed paths,

Conversely, any conserved quantity is defined as a function whose time derivative vanishes on the physical paths, that is a multiple ( derivatives of ) the equations of motion. This multiple defines the infinitesimal symmetry.


  • Symmetries of the equations of motion are not always symmetry of the effect. For example, the stretching is a symmetry of the equation of motion of a free particle, but not a symmetry of the Lagrangian with its effect. At such a symmetry of the equations of motion heard any conserved quantity.
  • The belonging to a symmetry conserved quantity as a function of time, place, and the velocities vanishes if and only if there is a gauge symmetry. In such a case, the motion equations are not independent, but an equation of motion is considered to be a result of the other. This implies the second Noethertheorem.
  • The Noethertheorem for translational and rotational movements Linear movements: The Noethertheorem explains why the speeds at time independent potential conservation of energy obtained when multiplying the Newtonian equations of motion: the velocity is the infinitesimal change in the place at a time shift.
  • Rotating Movements: Also explains the Noethertheorem why drehinvariantem potential at the product of the equations of motion leads to the cross product of the conservation of angular momentum in the direction: the cross product is the infinitesimal change of rotating about the axis. The Euler turbine equation applies the conservation of angular momentum to the interpretation of rotating machines (turbines ).
  • On displacements and rotations of the place the Lagrangian is strictly invariant, that is, the function vanishes. This is not true for temporal shift and transformation to a uniformly moving reference system. Under temporal shifts the action is invariant if the Lagrangian but not only depends on the location and the speed of time. Then the Lagrangian changes under temporal shifts with order. The associated conserved quantity is defined as the energy

Is it known how the energy depends on the velocity, this equation defines the Lagrangian fixed up to a level that is linear in the velocities and does not contribute to energy. Because the Lagrangian is broken down, for example, in units that are homogeneous of degree in the speed, then wear them with to energy. If so, then the Lagrangian

In particular, in Newtonian physics of energy from the kinetic energy, which is square in the speed, and the speed-independent potential energy. Therefore, the Lagrangian times the kinetic energy plus potential energy times. In relativistic physics in measurement systems with the Lagrangian and the energy of a free particle of mass