Noether theorem

The Noether theorem (formulated in 1918 by Emmy Noether ) links elementary physical quantities such as charge , energy and momentum with geometric properties, namely the invariance (immutability) of the effect under symmetry transformations :

Every continuous symmetry of a physical system has a conservation quantity .

A symmetry is a transformation (for example a rotation or displacement) that does not change the behavior of the physical system. The reverse also applies:

Every conserved quantity is a generator of a symmetry group.

A conservation quantity of a system of particles is a function of the time , the locations and the velocities of the particles, the value of which does not change in any orbit they traverse in the course of time . So the energy of a non-relativistic particle of mass moving in potential is a conserved quantity. This means that for every path that satisfies the equation of motion , the following applies at all times : ${\displaystyle t}$ ${\displaystyle x}$ ${\displaystyle v}$ ${\displaystyle x(t)}$ ${\displaystyle E(t,x,v)={\tfrac {1}{2}}\,m\,v^{2}+V(x)}$ ${\displaystyle m}$ ${\displaystyle V}$ ${\displaystyle x(t)}$ ${\displaystyle m{\tfrac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+\mathrm {grad} \,V(x)=0}$ ${\displaystyle t}$

${\displaystyle E{\bigl (}t,x(t),{\tfrac {\mathrm {d} x}{\mathrm {d} t}}(t){\bigr )}=E{\bigl (}0,x(0),{\tfrac {\mathrm {d} x}{\mathrm {d} t}}(0){\bigr )}}$.

Examples of symmetries and associated conservation quantities

• From the homogeneity of time (select Start Time does not matter) follows the conservation of energy ( energy conservation ). The energy of a pendulum remains the same if friction is neglected, but not the energy of a swing on which a child changes the length from the suspension to the center of gravity by lifting and lowering his body.
• From the homogeneity of the space (choice of the starting place does not matter) the conservation of the momentum results (momentum conservation law ). The momentum of a free particle is constant, but not the momentum of a particle in the sun's gravitational field ; their location is essential for the movement of the particle. Because a free particle mass is unchanged at a uniform speed moving, when looking at a uniformly moving observer, is the weighted starting location , a conserved quantity . Generalized to several particles, it follows that the center of gravity moves at a uniform speed when the total force disappears. ${\displaystyle m}$ ${\displaystyle v}$ ${\displaystyle S(t,x,v)=m\,(x-t\,v)}$ ${\displaystyle S(t,x(t),v(t))=m\,x(0)}$
• The isotropy of space, i.e. the rotational invariance (direction in space does not matter), results in the conservation of angular momentum ( angular momentum conservation law ). In this way, the angular momentum of a particle in the sun's gravitational field is retained, because the gravitational potential is the same in all directions. ${\displaystyle -G\,m\,M{\tfrac {1}{r}}}$

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

Mathematical formulation

effect

The relationship between symmetries and conservation quantities formulated in Noether's theorem applies to physical systems whose equations of motion or field can be derived from a principle of variation . One demands here that the so-called effect functional assumes an extreme value (see also principle of the smallest effect ).

With the movement of mass points this effect is functional through a Lagrangian function of time , place and speed ${\displaystyle S}$ ${\displaystyle t}$ ${\displaystyle x}$ ${\displaystyle {\dot {x}}={\tfrac {\mathrm {d} x}{\mathrm {d} t}}}$

${\displaystyle {\mathcal {L}}(t,x(t),{\dot {x}}(t))}$

characterizes and assigns the time integral to every differentiable trajectory ${\displaystyle x\colon t\mapsto x(t)}$

${\displaystyle S[x]=\int _{t_{1}}^{t_{2}}{\mathcal {L}}\left(t,x(t),{\dot {x}}(t)\right)\,\mathrm {d} t}$

to. For example, in Newtonian physics the Lagrangian function of a particle in potential is the difference between kinetic energy and potential energy : ${\displaystyle \Phi }$ ${\displaystyle T={\frac {m}{2}}v^{2}}$ ${\displaystyle V=\Phi q}$

${\displaystyle {\mathcal {L}}(t,x(t),{\dot {x}}(t))={\frac {1}{2}}\,m\,{\dot {x}}^{2}(t)-V(x(t))}$

The path actually traversed physically, which goes through the start point at the start and through the end point at the end , makes the value of the effect stationary (or extremal ) in comparison with all other (differentiable) paths that go through the same start or end point . The path actually physically traversed therefore fulfills the equation of motion ${\displaystyle t_{1}}$ ${\displaystyle x_{1}=x(t_{1})}$ ${\displaystyle t_{2}}$ ${\displaystyle x_{2}=x(t_{2})}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial }{\partial {\dot {x}}}}{\mathcal {L}}={\frac {\partial }{\partial x}}{\mathcal {L}}}$

(For derivation see the calculus of variations ). This corresponds precisely to Newton's equation of motion

${\displaystyle m{\ddot {x}}=-{\frac {\partial V}{\partial x}}(x)}$.

Differential equations that can be derived from an action functional through variation are called variationally self-adjoint. All elementary field and motion equations in physics are variationally self-adjoint.

symmetry

A differential equation is said to have symmetry if there is a transformation of the space of the curves that maps the solutions of the differential equations to solutions. Such a transformation is obtained for variationally self-adjoint differential equations if the transformation leaves the action functional invariant up to the boundary terms. The Noether theorem states that the invariance of the effect functional compared to a one-parameter continuous transformation group results in the existence of a conservation quantity and that, conversely, every conservation quantity results in the existence of an (at least infinitesimal) symmetry of the effect.

We limit ourselves here to symmetries in classical mechanics .

Is a one-parameter, differentiable group of transformations (sufficiently differentiable) curves on curves mapping, and belong to the parameter value to the identical picture . ${\displaystyle \Phi _{s}}$ ${\displaystyle \Gamma \colon t\mapsto x(t)}$ ${\displaystyle \Gamma _{s}\colon t\mapsto x(s,t,\Gamma )}$ ${\displaystyle s=0}$ ${\displaystyle \Phi _{0}\Gamma \colon t\mapsto x(t)}$

For example, each curve maps to the curve that was passed through earlier. The transformation with shifts each curve by a constant . ${\displaystyle \Phi _{s}\Gamma =\Gamma _{s}}$ ${\displaystyle \Gamma _{s}\colon t\mapsto x(t+s)}$ ${\displaystyle \Gamma }$ ${\displaystyle s}$ ${\displaystyle \Phi _{s}\Gamma =\Gamma _{s}}$ ${\displaystyle \Gamma _{s}\colon t\mapsto x(t)+s\,c}$ ${\displaystyle s\,c}$

The transformations are called local if the derivative is in the identical mapping, the infinitesimal transformation ${\displaystyle \Phi _{s}}$

${\displaystyle {\frac {\partial }{\partial s}}_{|_{s=0}}x(s,t,\Gamma )\ ,}$

for all curves as a function of time, place and speed , evaluated on the curve , can be written, ${\displaystyle \Gamma }$ ${\displaystyle \delta x(t,x,v)}$ ${\displaystyle x}$ ${\displaystyle v}$ ${\displaystyle \Gamma }$

${\displaystyle \delta x\left(t,x(t),{\frac {\mathrm {d} x}{\mathrm {d} t}}\right)={\frac {\partial }{\partial s}}_{|_{s=0}}x(s,t,\Gamma )}$.

For example, the shifts in time and place are local and belong to the infinitesimal transformation or to . ${\displaystyle \delta x=v}$ ${\displaystyle \delta x=c}$

Now be the Lagrangian of the mechanical system. Then the local transformations are called symmetries of action if the Lagrangian for all curves changes only by the time derivative of a function , evaluated for infinitesimal transformations : ${\displaystyle {\mathcal {L}}(t,x,v)}$ ${\displaystyle \Phi _{s}}$ ${\displaystyle \Gamma }$ ${\displaystyle K(t,x)}$ ${\displaystyle \Gamma }$

${\displaystyle {\frac {\partial }{\partial s}}_{|_{s=0}}{\mathcal {L}}\left(t,x(s,t),{\frac {\partial x}{\partial t}}(s,t)\right)={\frac {\mathrm {d} }{\mathrm {d} t}}K\left(t,x(t)\right)}$

Because then the effect only changes by boundary terms

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} s}}S[\Gamma _{s}]_{|_{s=0}}&=\int _{t_{1}}^{t_{2}}\mathrm {d} t\,{\frac {\partial }{\partial s}}_{|_{s=0}}{\mathcal {L}}\left(t,x(s,t),{\frac {\partial x}{\partial t}}(s,t)\right)\\&=\int _{t_{1}}^{t_{2}}\mathrm {d} t\,{\frac {\mathrm {d} }{\mathrm {d} t}}K\left(t,x(t)\right)=K\left(t_{2},x(t_{2})\right)-K\left(t_{1},x(t_{1})\right)\end{aligned}}}$.

The connection between this definition of the symmetry of the effect and the conserved quantity becomes clear when one carries out the partial derivatives of the Lagrangian and uses the definition of the infinitesimal transformation as a shorthand ${\displaystyle s}$

${\displaystyle {\begin{aligned}{\frac {\partial }{\partial s}}_{|_{s=0}}{\mathcal {L}}(t,x(s,t),{\frac {\partial x}{\partial t}}(s,t))&={\frac {\partial x(s,t)}{\partial s}}_{|_{s=0}}{\frac {\partial }{\partial x}}{\mathcal {L}}(t,x,v)+{\frac {\partial ^{2}x(s,t)}{\partial s\partial t}}_{|_{s=0}}{\frac {\partial }{\partial v}}{\mathcal {L}}(t,x,v)\\&=\delta x{\frac {\partial }{\partial x}}{\mathcal {L}}(t,x,v)+{\frac {\mathrm {d} \delta x}{\mathrm {d} t}}{\frac {\partial }{\partial v}}{\mathcal {L}}(t,x,v)\end{aligned}}}$

If you add the first term to a multiple of the equation of motion and subtract the additions from the second term, the result is

${\displaystyle {\frac {\partial }{\partial s}}_{|_{s=0}}{\mathcal {L}}\left(t,x(s,t),{\frac {\partial x}{\partial t}}(s,t)\right)=\delta x\left({\frac {\partial }{\partial x}}{\mathcal {L}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial }{\partial v}}{\mathcal {L}}\right)+{\frac {\mathrm {d} }{\mathrm {d} t}}\left(\delta x{\frac {\partial }{\partial v}}{\mathcal {L}}\right)}$

and the equation of definition of an infinitesimal symmetry of an effect is

${\displaystyle \delta x\left({\frac {\partial }{\partial x}}{\mathcal {L}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial }{\partial v}}{\mathcal {L}}\right)+{\frac {\mathrm {d} }{\mathrm {d} t}}\left(\delta x{\frac {\partial }{\partial v}}{\mathcal {L}}-K\right)=0}$.

But since the factor of the equations of motion disappears on the physically traversed paths, this equation says that the function ${\displaystyle \delta x}$

${\displaystyle Q=\delta x{\frac {\partial }{\partial v}}{\mathcal {L}}-K}$,

the noether charge associated with symmetry does not change on the physically traversed paths:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}Q\left(t,x_{\mathrm {phys} }(t),{\frac {\mathrm {d} x_{\mathrm {phys} }(t)}{\mathrm {d} t}}\right)=0}$

Conversely, every conservation quantity is, by definition, a function whose time derivative vanishes on physical paths, i.e. it is a multiple (of derivatives) of the equations of motion. This multiple defines the infinitesimal symmetry . ${\displaystyle Q(t,x,v)}$ ${\displaystyle \delta x}$

Remarks

• Symmetries of the equations of motion are not always symmetries of effect. For example, the stretching is a symmetry of the equation of motion of the free particle, but not a symmetry of its effect with the Lagrangian function . No conserved quantity belongs to such a symmetry of the equations of motion. ${\displaystyle x(t)\mapsto \mathrm {e} ^{s}x(t)}$ ${\displaystyle m{\ddot {x}}=0}$ ${\displaystyle {\mathcal {L}}={\tfrac {1}{2}}\,m\,v^{2}}$
• The conservation quantity belonging to a symmetry as a function of time, place and speed disappears exactly when it is a question of a calibration symmetry . In such a case the equations of motion are not independent, but one equation of motion counts as a consequence of the others. This says the second noether theorem.
• The noether theorem for translational and rotating movements
  • Translational movements : The Noether theorem explains why one obtains the law of conservation of energy when multiplying Newton's equations of motion with the velocities with time-independent potential : The velocity is the infinitesimal change in the location with a time shift.
  • Rotating movements : The Noether theorem also explains why the product of the equations of motion with the cross product leads to the conservation of angular momentum in the direction of rotation invariant potential : The cross product is the infinitesimal change of when rotating around the axis . The Euler turbine equation applies to the conservation of angular momentum to the interpretation of rotating machines (turbines). ${\displaystyle {\vec {n}}\times {\vec {x}}}$ ${\displaystyle {\vec {n}}}$ ${\displaystyle {\vec {n}}\times {\vec {x}}}$ ${\displaystyle x}$ ${\displaystyle {\vec {n}}}$
• The Lagrangian is strictly invariant in the case of shifts and rotations of the location, i.e. the function disappears. However, this does not apply to time shifts and to transformation to a uniformly moving reference system . The effect is invariant under temporal shifts if the Lagrangian only depends on the location and the speed , but not on the time. Then the Lagrangian changes under time shifts in order to . The associated conservation quantity is by definition the energy ${\displaystyle K}$ ${\displaystyle x}$ ${\displaystyle v}$ ${\displaystyle {\tfrac {\mathrm {d} }{\mathrm {d} t}}K}$ ${\displaystyle K={\mathcal {L}}}$

${\displaystyle E=v{\frac {\partial {\mathcal {L}}}{\partial v}}-{\mathcal {L}}}$.

If it is known how the energy depends on the speed, then this equation defines the Lagrangian up to a proportion that is linear in the speeds and does not contribute to the energy. Because if you break down the Lagrangian function, for example, into parts that are homogeneous in terms of speed, then they contribute to the energy. So is , then is the Lagrangian ${\displaystyle {\mathcal {L}}_{n}v^{n}}$ ${\displaystyle n}$ ${\displaystyle v{\tfrac {\partial }{\partial v}}{\mathcal {L}}_{n}v^{n}-{\mathcal {L}}_{n}v^{n}=(n-1){\mathcal {L}}_{n}v^{n}}$ ${\displaystyle \textstyle E=\sum _{n}E_{n}(x)v^{n}\,}$

${\displaystyle {\mathcal {L}}=\sum _{n}{\frac {1}{n-1}}E_{n}v^{n}}$.

In particular is to Newtonian physics, the energy from the kinetic energy is quadratic in speed, and the speed-independent potential energy . Therefore the Lagrangian is times the kinetic energy plus times the potential energy. In relativistic physics applies in measuring systems with a free particle of mass for the Lagrangian and energy : ${\displaystyle n=2}$ ${\displaystyle n=0}$ ${\displaystyle {\tfrac {1}{2-1}}}$ ${\displaystyle {\tfrac {1}{0-1}}}$ ${\displaystyle c=1}$ ${\displaystyle m}$

${\displaystyle {\mathcal {L}}=-m{\sqrt {1-v^{2}}}}$

${\displaystyle E={\frac {m}{\sqrt {1-v^{2}}}}}$

Individual evidence

1. ↑ Eugene J. Saletan and Alan H. Cromer: Theoretical Mechanics . John Wiley & Sons, 1971, ISBN 0-471-74986-9 , pp. 83-86 (English).

literature

• E. Noether: Invariants of arbitrary differential expressions. In: God. Nachr. 1918, pp. 37-44. Summary in Zentralblatt MATH.
• E. Noether: Invariant Variation Problems . In: God. Msg. 1918, pp 235-257. Summary in Zentralblatt MATH.