The thing starts with invariance, conservation and symmetry…
Then the whole consideration moves on to the functionals…
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:
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,
*** A reminder for the relation between the Hamiltonian H and the Lagrangian L,
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.