The thing starts with **invariance**, **conservation** and **symmetry**…

**Invariant** is something that is unchanged by a coordinate transformation.

**Conserved** is something that is unchanged by a process within a given coordinate system.

**Symmetry** could be considered a broader property of some ‘universe’ in which some entity of interest remains the same under a specific operation. To measure symmetry one needs to apply some transformation and see if there is a change in the object of interest.

In addition, **theories of relativity** are being built on entities that are supposed to be invariant among various coordinate systems.

Then the whole consideration moves on to the **functionals**…

A **functional** is a ‘mathematical tool’ that takes functions and maps them into real numbers.

This is indeed the work of **definite integrals**, so usually functionals are expressed as definite integrals. A typical formulation is as follows:

where *L* is the *Lagrangian* of the functional *J* and the arguments in *L(.)* are the independent variable, the dependent variables and their derivatives. In this definition, *the Lagrangian is the integrand of the functional*.

A simple example of a functional is that of the ‘distance’. The distance on a path on a plane *xy* between points *a, b*, which is described by a function *y=y(x)*, can be defined as the integral of all infinitesimal lengths *ds* as follows:

assuming that

exists in the closed interval *[a,b]*.

A *geodesic* is nothing more than the shortest such path constrained to the same surface (the plain in this example). Since we talk about a “shortest” this means we need to consider *extremals*. Here we find expressions like the celebrated *Fermat’s principle*, which requires that a trajectory’s time interval be minimum between the end-points,

or the equally celebrated *Hamilton’s principle*, which requires that a trajectory be the shortest between two times *a, b*,

*K-U* being the well known Kinetic-Potential energy difference.

Since we are dealing with functionals and need to ‘make them’ extremals, the **Euler-Lagrange equation** comes to play. Since,

the *x ^{μ}(t)* that make

*J*an extremum are the solutions of the

**,**

*N*Euler-Lagrange equations
*Conservation laws and (corresponding) symmetries* arise as consequences of the Euler-Lagrange equations either as they are, or by including momentum. If *L* remains unchanged by, say, a translation, the equations result a zero and momentum becomes constant, thus conserved; that’s one symmetry. On the other side, if *L* remains unchanged in time then the corresponding Hamiltonian becomes constant, thus energy is preserved; that’s another symmetry.

*** A reminder for the relation between the Hamiltonian *H* and the Lagrangian *L*,

What **Emmy Noether** did in 1918 was to *interconnect extremal functionals with invariance under infinitesimal transformations* in her famous Theorem, which is considered to be largely influential in modern physics during the 20th century. Her theorem states in words that *if a functional is extremal and it is invariant under an infinitesimal transformation then a conservation law holds.*

A consequence of this theorem was the discovery of conservation laws that could not be derived from Euler-Lagrange equations, like the conservation of both energy and momentum in cases in which neither of them is separately conserved!

*** The featured image is Google’s Emmy Noether’s 133th birthday doodle.