Higher order mechanics on graded bundles

December 9th, 2014 by ajb
In collaboration with K. Grabowska and J. Grabowski, we have applied the recently discovered notion of a weighted algebroid to mechanics on graded bundles[1].

In our preprint “Higher order mechanics on graded bundles” We present the corresponding Tulczyjew triple for this situation and derive the phase equations from an arbitrary (maybe singular) Lagrangian or Hamiltonian, as well as the Euler-Lagrange equations. This is all done essentially in the first order set-up of mechanics on a Lie algebroid subject to vakonomic constraints. The amazing this is that the underlying graded bundle structure gives this whole picture the flavour of higher derivative mechanics. Within this framework we recover classical higher order mechanics, but we can study some more exotic situations.

For example, we geometrically derive the (reduced) higher order Euler-Lagrange equations for invariant higher order Lagrangians on Lie groupoids. To our knowledge, not much work has been done in understanding such systems [2,3]. We hope that the example on Lie groupoids turns out to be useful, maybe in say control theory.

[1] A.J. Bruce, K. Grabowska & J. Grabowski, Linear duals of graded bundles and higher analogues of (Lie) algebroids, arXiv:1409.0439 [math-ph], (2014).

[2] L. Colombo & D.M. de Diego, Lagrangian submanifolds generating second-order Lagrangian mechanics on Lie algebroids, XV winter meeting of geometry, mechanics and control, Universidad de Zaragoza, (2013). http://andres.unizar.es/ ei/2013/Contribuciones/LeoColombo.pdf

[3] M. Jozwikowski & M. Rotkiewicz, Prototypes of higher algebroids with applications to variational calculus, arXiv:1306.3379v2 [math.DG] (2014).

Breakthrough Prize in Fundamental Physics Symposium Videos

December 5th, 2014 by ajb
Higgs event The videos from the Breakthrough Prize in Fundamental Physics Symposium are now available to watch, follow the link below. The symposium was held on the 10th November at Stanford University and co-hosted by UC-San Francisco and UC-Berkeley.

There was a panel discussion with Adam Riess, Brian Schmidt, Saul Perlmutter and Yuri Milner, and individual 20-minute talks from Nima Arkani-Hamed, Juan Maldacena, Andrei Linde, Stephen Shenker, Alexei Kitaev, Patrick Hayden, John Preskill, Nathan Seiberg, Joe Polchinski and Uros Seljak.

Breakthrough Prize in Fundamental Physics Videos 2015

My Birthday

November 23rd, 2014 by ajb
cake I did not realise this until today, but I share my birthday with John Wallis (23 November 1616 – 28 October 1703). Wallis made contributions to infinitesimal calculus, analytic geometry, algebra and the theory of colliding bodies. He is best known for the infinity symbol $latex \infty$.

Wallis was also a code breaker and used his mathematical skills during the Civil war to decode Royalist messages for the Parliamentarians.

John Wallis Wikipedia.

Engineers should embrace their artistic side.

November 22nd, 2014 by ajb
? More warnings about a shortage of STEM graduates. Where are these messages coming from? There is no shortage, many graduates in engineering for example are forced to work in jobs that are nothing to do with their degrees, or even worse they are in non-graduate jobs.

Engineering needs to emphasise its creative side to encourage more young people to take it up as a career.

Sir John O’Reilly

This is fine. The fields of science and engineering do require some creative thinking. There is an element are art to this.

But I am worried about the continual message of a lack of STEM graduates where in reality many are unemployed. I really don’t understand the B.S. here.

‘Embrace engineering’s creative side’ to fix skills crisis

Poor maths skills in Welsh Schools

November 13th, 2014 by ajb
flag of wales Schools watchdog Estyn says too many pupils struggle with the basics of mathematics and maths skills were “at best, average” in more than half of schools inspected in Wales last year.

The mathematics skills and the teaching methodologies in Welsh schools seems to be a continuing source of worry.

The attitude that it is okay to not be good at basic mathematics needs to change. There is some social acceptability in being poor in mathematics. I am sure that when English teachers, lets say at a party, explain that they are English teachers are not met with a funny look and the response “I hated English at school and I cannot read or write”…

Concerns over pupils’ maths skills at half of Welsh schools BBc News

Why there is no equivalence principle for electromagnetic theory

November 10th, 2014 by ajb
Quite regularly one will come across a website, blog or some thread on a forum that says the gravity is just electromagnetism. For sure they are not the same. However, I am not sure what would constitute the ‘nail in the coffin’ for all these clams.

With this in mind, I am not going to try to debunk every such claim. However, I think the reason for this seeming equivalence comes from the static limit. In this limit it is true that there are many formal similarities between Newtonian gravity and electrostatics. In fact there are many formal similarities in the full classical theories, both are very geometrical in nature, but I won’t go into details here.

I just wanted to point out one very clear difference between gravity and electromagnetism that can be seen in this static limit. That is the lack of a generalisation of the equivalence principal for electromagnetic theory. This principal in gravity is very important and one that I will comment on in due course.

The static limit
I am guessing that we have all seen Coulombs law for electrostatics and Newton’s law for gravity. Let me just write them down

$latex F = k \frac{qQ}{r^{2}}$,

where $latex k$ is Coulombs constant, it is a measure of the strength of the electrostatic force and $latex q$ and $latex Q$ are the electric charges of two point particles. The above expression is the electrostatic force between two such charged particles.

Similarly we have Newton’s law of gravity

$latex F = G \frac{mM}{r^{2}}$,

where $latex G$ is Newton’s constant which measures the strength of the gravitational force and $latex m$ and $latex M$ are the masses of two point particles. The above expression measures the gravitational force between these particles.

These expressions for the forces should be seen as the static non-relativistic limit. I just mean that as long as the particles are moving slow enough then the change in the fields can be viewed as instantaneous. This is okay for many applications, but it is not the full picture. However, it is the one you see at high school.

The formal similarities at this level are clear. You just need to swap constants and interchange charge and mass. But this does not mean they are the same, and there is a subtle issue here. Before that we need Newton’s law of motion

Newton’s second law
Newton’s second law tells us that the force exerted on a particle is proportional to the acceleration of that particle. Moreover, the constant of proportionality is the (inertial) mass.

$latex F = m a $.

That is all we will need.

The gravitational equivalence principal
Let us think of the particle of mass $latex m$ as a test particle. That is we will think of how it is moving in the gravitational field generated by the particle $latex M$ and that it does not generate a gravitational field of its own. This approximation is good for small objects moving in the gravitational field of big objects; say planets around a star or satellites in orbit around the Earth.

Now we can examine how the small mass is influenced by the big mass. We should just equate the two expressions due to Newton

$latex ma = G \frac{mM}{r^{2}} $,

for which we can solve for the acceleration

$latex a = G \frac{M}{r^{2}}$.

We notice a very amazing thing. The small mass cancels from both sides of the equation. (We assume that gravitational mass and inertial mass are the same. This seems to be case in nature.)

This means that the motion of the test particle does not care about any of the intrinsic properties of that particle. The only things it does care about is the initial position and velocity. This is one form of the equivalence principal which has it’s roots in the experimental work of Galileo- acceleration of a test particle due to gravity is independent of the mass being accelerated.

Thus, really all the information about the test particles motion is encoded in the gravitational field alone. All test particles whatever their mass will behave the same. This is the clue that gravity can be formulated very intrinsically in terms of space-time geometry alone; this leads to general relativity which is not the subject for today.

The electromagnetic version
Now let us play the same game with electrostatics…

$latex ma = k \frac{q Q}{r^{2}} $,

where we think of the test particle $latex (m, q)$ moving in the electric field generated by the particle $latex (M,Q)$. Now solving for the acceleration gives us

$latex a = k \left(\frac{q}{m}\right) \frac{Q}{r^{2}} $.

Now we see the difference. The motion of the test particle does depend on the intrinsic properties of that particle, namely the charge-mass ratio. There is no similar statement like for gravitational physics; there is no equivalence principle.

Closing remarks
Everything above is done in a particular limit of the full classical theories. The same picture is true when we examine the motion of test particles in general relativity and the Lorentz force in electromagnetic theory. I have chosen these limits because I think this is clear and also the source of the instance that gravity is just electromagnetic theory. I have pointed out one clear and explicit difference.

One can do the same with Coulomb’s law for the magnetic force. Although magnetism is a bit more complicated we can examine the situation for point-like poles. This is okay for small enough poles that are well separated. You will reach the same conclusion that there is no equivalence principal in this situation. Thus, gravity is not magnetism either.

Talk for Polish physics students

November 5th, 2014 by ajb
On Friday 7th November I will be giving a short talk to physics students entitled ‘Fermions in Physics: from anticommuting variables to supermanifolds’, as part of the Ogólnopolska Sesja Kół Naukowych Fizyków.

This translates as ‘Scientific Session of the Nationwide Circle of Physicists’. This year the session is in the Tricity area.

I hope to tell them a little about the passage from the canonical anticommutation relations to Grassmann algebras and then supermanifolds. I have 45mins to get them interested in this geometric side of mathematical physics.

I will post the slides here after the event, but they may not make much sense without me telling this you this story.

UK may fall behind in science

October 16th, 2014 by ajb
mad There is always some worry that government cut backs will effect the status of UK science. The latest warning comes from the Campaign for Science and Engineering (CaSE) which represents 100 scientific bodies and businesses.

If you look at what other countries are doing they are investing heavily. They are all going upward. Unfortunately the UK’s (investment) is going downwards.

Dr Sarah Main Director, Case speaking to the BBC.

The UK government has invested less in science than the European Union average for many years. The EU average rose from 2.04% of GDP in 2011 to 2.07% of GDP in 2012. The UK investment dropped from 1.78% to 1.72% of GDP.

This makes me worry about the UK loosing its place in science, though like make things for a small country we punch above our weight. I just hope we can continue to do so in science. Moreover, this downwards trend on funding is worrying for science students and especially PhD students. The situation with postdoc positions is difficult enough without further erosion of UK science.

UK science ‘losing ground’ to rival nations, BBC NEWS

Campaign for Science and Engineering

How English became the language of science

October 13th, 2014 by ajb
English today is the main language of science across the globe. It has not always been like that, a mixture of English, French and German in about equal proportions was the norm in the 19th century. Before that Latin was the language used across western Europe; from the late medieval period to the mid-17th Century.

It may be no surprise that German was cast out as a common scientific language after the first world war. It is not that English rose to the top of the pile, but rather than German was cut down. Then after the second world war the United States of America became a leading scientific power and that for sure influenced the choice of international language.

Almost of today’s scientific discoveries will be published in English. Moreover, English seems to be the language that scientists who do not share a common mother tongue will use. For example, here in Warsaw international visitors will tend to use English when conversing with Polish researchers.

You can find out more details about the rise on English via the link below.

Nobel Prize: How English beat German as language of science BBC News Magazine

How to avoid writers block

October 11th, 2014 by ajb
Caravaggio, 1605-1606. I don’t find it at all easy to write papers or lectures notes, even if the hard work of getting all the material together has been done. To help with the procrastination I may try out the ideas of the writers linked to below.

I am not sure exactly how my fellow mathematicians sharing the office will react though…

7 Famous Writers That Worked In The Nude