Noether’s theorem establishes a relationship between continuous symmetries and conservation laws in systems described by an action principle. In its standard form, every differentiable, one-parameter variational symmetry produces a quantity conserved when the equations of motion hold. Published by Emmy Noether in 1918, it explains why time translations, spatial translations, and rotations are associated with conservation of energy, momentum, and angular momentum. The name usually denotes her first theorem, distinguished from a second theorem concerning symmetries involving arbitrary functions. (arxiv.org)
Historical and mathematical setting
Noether presented her results in Invariante Variationsprobleme (“Invariant Variation Problems”). The paper combined the calculus of variations with the theory of continuous transformation groups, now expressed through Lie groups. Rather than deriving a separate conservation law for each physical model, it established general relations between invariance properties of variational problems and their associated differential equations. Its treatment included both finite-parameter symmetry groups and groups depending on arbitrary functions. (arxiv.org)
The work addressed questions arising in general relativity, particularly the status of energy conservation in generally covariant theories. This setting helped motivate the distinction between symmetries governed by finitely many parameters and local transformation freedom. The two theorems therefore answer related but different questions: the first constructs conserved currents, while the second establishes identities among equations of motion. (arxiv.org)
Statement in classical mechanics
In classical mechanics, consider coordinates (q_i(t)) and an action [ S[q]=\int_{t_1}^{t_2}L(q,\dot q,t),dt, ] where (L) is the Lagrangian. Stationarity of the action gives the Euler–Lagrange equations [ \frac{d}{dt}\frac{\partial L}{\partial\dot q_i} -\frac{\partial L}{\partial q_i}=0. ] A continuous transformation is a variational symmetry when it preserves the action’s variational structure; exact invariance of the Lagrangian is sufficient, but not necessary. A change by a total time derivative is also allowed, because it contributes only an endpoint term. (damtp.cam.ac.uk)
For a fixed-time infinitesimal transformation [ \delta q_i=\epsilon,\xi_i(q,t), \qquad \delta L=\epsilon,\frac{dB}{dt}, ] the conserved quantity is [ Q=\sum_i p_i\xi_i-B, \qquad p_i=\frac{\partial L}{\partial\dot q_i}. ] Along solutions, (dQ/dt=0). The proof substitutes the equations of motion into the variation of (L), converting that variation into a total derivative. Transformations that also change time require a corresponding time-transformation term in the charge. (damtp.cam.ac.uk)
Familiar conservation laws
The theorem organizes several standard correspondences:
- Time translation: If (L) has no explicit time dependence, the conserved quantity is [ E=\sum_i p_i\dot q_i-L. ] For ordinary mechanical systems, this is energy.
- Spatial translation: Invariance under displacement along a direction yields conservation of the corresponding component of momentum.
- Spatial rotation: Rotational invariance yields conservation of angular momentum; invariance about a single axis conserves the component along that axis. (damtp.cam.ac.uk)
For example, a particle with [ L=\tfrac12m\dot{\mathbf r}^{,2}-V(|\mathbf r|) ] has rotational symmetry because its potential energy depends only on distance from the origin. Its angular momentum (\mathbf r\times m\dot{\mathbf r}) is therefore conserved. A fixed central potential generally lacks spatial translation symmetry, so the particle’s linear momentum need not be conserved. The relevant symmetry belongs to the complete dynamical model, including external potentials, rather than merely to its kinetic term. (damtp.cam.ac.uk)
Field theories and local conservation
In field theory, the classical action is written [ S=\int d^dx,\mathcal L(\phi_a,\partial_\mu\phi_a,x). ] For a fixed-coordinate variation (\delta\phi_a=\epsilon\Delta\phi_a) satisfying (\delta\mathcal L=\epsilon\partial_\mu K^\mu), the associated Noether current is [ j^\mu= \sum_a\frac{\partial\mathcal L} {\partial(\partial_\mu\phi_a)}\Delta\phi_a-K^\mu. ] The field equations imply the continuity equation (\partial_\mu j^\mu=0). (damtp.cam.ac.uk)
This equation expresses local conservation: changes within a region are balanced by current crossing its boundary. The integrated charge [ Q=\int d^{d-1}x,j^0 ] is constant when the boundary flux vanishes. Translations in spacetime produce the energy–momentum tensor. Internal symmetries, such as a uniform phase rotation of a complex field, produce other charges; in charged matter theories, the relevant phase symmetry is associated with electric charge. (damtp.cam.ac.uk)
The second theorem and gauge freedom
Noether’s second theorem concerns variational symmetries whose parameters are arbitrary functions of position and time. Such symmetries imply differential identities among the Euler–Lagrange expressions, valid even before imposing the equations of motion. In gauge theories, these identities reflect redundancy in the description and explain why some field equations are not independent. (arxiv.org)
Consequently, a local gauge symmetry does not supply an independent physical conserved quantity for every arbitrary parameter function. Its relationship with global symmetries, constraints, and boundary charges requires distinguishing the first theorem’s conservation statements from the second theorem’s identities. Electromagnetism provides a central example of this distinction. (arxiv.org)
Scope and qualifications
The standard theorem requires a continuous variational symmetry, not simply a transformation preserving the equations of motion. Discrete transformations lack the infinitesimal parameter used in its construction and do not automatically produce a Noether current. Likewise, local conservation alone does not establish a conserved integrated charge without appropriate boundary conditions. (damtp.cam.ac.uk)
In quantum theory, a classical symmetry can fail to survive quantization. Such a quantum anomaly can modify the conservation equation of a classically conserved current. In general relativity, covariant conservation of stress-energy also does not by itself guarantee a globally conserved total energy: additional spacetime symmetries or suitable boundary structure are needed. (damtp.cam.ac.uk)