The logo for the AST

In 2008 the university where I took all my degrees, University of Rome Sapienza, due to gigantism, decided to split into 5 confederate universities. Among the fragments resulting from this schism, the one where my faculty happened to be, the AST (Athenaeum of Science and Technology), published a competition for the creation of its representative logo. The logo should have represented the faculties of which the AST is composed: Mathematical, Physical and Natural Sciences, Philosophy, Psychology, Engineering, Statistics and Aerospace sciences.

I decided to compete.

The theme I chose was the “Ouroboros“, the snake seizing its own tail… an ancient symbol that represents the eternal return (a concept dear to philosophers and mathematicians). I used this symbol to remember  about an episode in the history of science: the discovery by the german chemist Friedrich August Kekulé of the ring structure of the benzene molecule. His discovery was inspired by a dream in which he saw a snake eating its own tail. This episode, for me, linked psychology (the dream), phylosophy (the Ouroboros) and science.

So I put the 6-fold (incidentally, the AST has exaclty 6 faculties) molecule of benzene on the logo, with an Ouroboros winded upon it.

This is the result:

And this is the black and white version:

Eventually, my work didn’t win, it wasn’t considered sufficient. Neither was any other work, so the competition was cancelled.

I spoke with one of the members of the committee, asking him what was the problem with my artwork, and he told me that it was too bombastic, like the emblem of an American institution. He wanted something more easily recognizable, like an advertising logo… didn’t know we were selling some product.

But, since my motivation was strong, coming from a noble attachment to money, when the competition was opened again I wanted to participate again. This time I joined with a friend, trying together to think like a commercial artist. The result was this:

And the black and wite version:

Please don’t ask me what it represents, because it has no more significance than Nike’s swoosh (well, obviously there is the magnifying glass, which should recall some archetypes connected with research).

Again, we’ve lost. But this time the first and second prizes were assigned (these works are temporarily accessible at this page)… so hail to the winners.

In conclusion, the two logos I and my friend have made are now fluctuating in a limbo, threatened by oblivion, hoping to find a new assignment. If you’re aware of any institution looking for an emblem, please tell me.

P.S. I’ll be happy to change the letters accordingly.

F. Mercati

Noether’s second theorem – Part I

In her most famous (among physicists) paper [1], the Artemisia Gentileschi of mathematics, Emmy Noether, did not only prove the theorem which had probably the greatest influence over the twentieth century physics, but also proved one which had probably one of the smallest. The first is obviously the one best known as Noether’s theorem, but some people refers to it as Noether’s first theorem, to distinguish it from the theorem we are going to talk about today.

The theorem

Consider fields labeled with the subscript , whose dynamics is described by the action funcional:

which is invariant under a local transformation of the fields, written in infinitesimal form as:

where we assumed that the transformation depends only on the functions and their first derivatives . The variation of the lagrangian under the above transformation is (sum over repeated indexes is understood, also for the and index):

factorizing the functions and its first and second derivatives, with some integrations by parts, one obtains:

the invariance of the lagrangian under a generic local transformation (defined by an arbitrary choice of the functions ) imply the following constraints (the last of the three equations follows from the symmetry of with respect to the indices , ):

Let’s rationally rearrange the above equations. If we call:

we have that , and the second equation becomes:

so that it express the conservation of the current , since:

and, from the antisymmetry of we have , so that:

It is noteworthy that this conservation law we found does not apply only on solutions of the Euler-Lagrange equations (some people would say that it holds off-shell).
The first equation of the trio can be rephrased in a more convenient way:

and from the equation for :

so that we obtain an equation in terms only of the Euler-Lagrange expressions :

The last equation express the main content of the theorem: it enforces the role of redundancies played by the symmetries. In fact the physical content of the theory, more than in the lagrangian, is contained in the equations of motion:

they simply state that for every field its corresponding Euler-Lagrange expression vanishes on the solutions (or on-shell). There are of these equations. But the equation before establishes relations (recall that ) one for each independent local symmetry between the Euler-Lagrange expressions. So there are only independent equations of motion.

References

[1] E. Noether’s 1919 paper: arXiv:physics/0503066v1 (in 1919 the arXiv didn’t have hep-th… thanks to M. A. Tavel for the translation)

F. Mercati

The blog is open

This blog is officially open, check out the blog description to get an idea of the topics of this blog, and the who’s who to discover who are the authors.