Informal statement of the theoremAll fine technical points aside, Noether's theorem can be stated informally
If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time. A more sophisticated version of the theorem states that:
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. The conservation law of a physical quantity is usually expressed as a continuity equation.

The formal proof of the theorem uses only the condition of invariance to derive an expression for a current associated with a conserved physical quantity. The conserved quantity is called the Noether charge and the flow carrying that 'charge' is called the Noether current. The Noether current is defined up to a solenoidal vector field.

Historical contextMain articles:

Constant of motion, conservation law, and conserved currentA conservation law states that some quantity X describing a system remains constant throughout its motion; expressed mathematically, the rate of change of X (its derivative with respect to time) is zero:
Such quantities are said to be conserved; they are often called constants of motion, although motion per se need not be involved, just evolution in time. For example, if the energy of a system is conserved, its energy is constant at all times, which imposes a constraint on the system's motion and may help to solve for it. Aside from the insight that such constants of motion give into the nature of a system, they are a useful calculational tool; for example, an approximate solution can be corrected by finding the nearest state that satisfies the necessary conservation laws.

The earliest constants of motion discovered were momentum and energy, which were proposed in the 17th century by René Descartes and Gottfried Leibniz on the basis of collision experiments, and refined by subsequent researchers. Isaac Newton was the first to enunciate the conservation of momentum in its modern form, and showed that it was a consequence of Newton's third law; interestingly, conservation of momentum still holds even in situations when Newton's third law is incorrect.

Modern physics has revealed that the conservation laws of momentum and energy are only approximately true, but their modern refinements – the conservation of four-momentum in special relativity and the zero divergence of the stress-energy tensor in general relativity – are rigorously true within the limits of those theories. The conservation of angular momentum, a generalization to rotating rigid bodies, likewise holds in modern physics. Another important conserved quantity, discovered in studies of the celestial mechanics of astronomical bodies, was the Laplace-Runge-Lenz vector.

In the late 18th and early 19th centuries, physicists developed more systematic methods for discovering conserved quantities. A major advance came in 1788 with the development of Lagrangian mechanics, which is related to the principle of least action. In this approach, the state of the system can be described by any type of generalized coordinates q; the laws of motion need not be expressed in a Cartesian coordinate system, as was customary in Newtonian mechanics.

The action is defined as the time integral I of a function known as the Lagrangian L
where the dot over q signifies the rate of change of the coordinates q
Hamilton's principle states that the physical path q(t) – the one truly taken by the system – is a path for which infinitesimal variations in that path cause no change in I, at least up to first order. This principle results in the Euler–Lagrange equations.

Thus, if one of the coordinates, say qk, does not appear in the Lagrangian, the right-hand side of the equation is zero, and the left-hand side shows that,

where the conserved momentum pk is defined as the left-hand quantity in parentheses. The absence of the coordinate qk from the Lagrangian implies that the Lagrangian is unaffected by changes or transformations of qk; the Lagrangian is invariant, and is said to exhibit a kind of symmetry. This is the seed idea from which Noether's theorem was born.

Several alternative methods for finding conserved quantities were developed in the 19th century, especially by William Rowan Hamilton. For example, he developed a theory of canonical transformations that allowed researchers to change coordinates so that coordinates disappeared from the Lagrangian, resulting in conserved quantities. Another approach and perhaps the most efficient for finding conserved quantities is the Hamilton-Jacobi equation.