New project (Creation of the physical world)

A new idea or approach, rather, to understand the world: that is to create the world from scratch. Questions like what is photons,do quarks really exist etc will hopefully be answered. To truly understand nature without accepting blindly what is being offered by scientists.

To make a start, I am going to assume a few things. Those assumptions will be put into a list which I haven’t started to think of. With the hope of understanding really how this world works and what nature really is, the only way is through logical reasoning. Experimental results will be used as a guide.

One (major) assumption that I can think of now is Mathematics as the language of nature. Does that mean everything that the Maths allows implies a correspondent physical phenomena? I do not know.

The geometry of Principal Bundles

I am writing this post as a result of the process of internalizing principal bundles and some geometrical structures.

Geometry is usually associated with some structures that allow one to compare some properties of 2 points. For example, distance between the points, or directions, angle between them etc. For non Euclidean(not flat) manifold or geometry, to compare 2 vectors at 2 points amounts to bringing one of the vectors to the other point. To bring or transport the vector from one point to the other, an additional structure is required. This is known as the connection.

Above is what parallel transport is about for normal space. What is the meaning of parallelism for principal fibre bundles? The rough idea is that we have a curve in base manifold and parallelism is that we have a corresponding curve in total or bundle space. In such a way that the projection of the curve in total space gives the curve in base manifold.

The next question is then how to find this curve in total space? We need to define a corresponding connection for principal fibre bundles. There are 2 equivalent definitions of connection for principal fibre bundles. The first is that it is a collection of horizontal subspace such that the horizontal vector at the fibre corresponds to tangent vector at base manifold under the projection.

However, this definition is not quite useful as there is no instructions on how to obtain these horizontal vectors. Hence the second definition. Connection on a principal fibre bundle is a 1-form with values in the Lie algebra of G. Why do we defined it this way? Vertical subspace is tangent space to fibre. But fibre is isomorphic to structure group G hence there is a correspondence between vertical subspace and tangent spaces of G. Hence the connection 1-form will act on vectors giving vectors parallel to fibres. This implies that connection 1-form acting on horizontal vectors gives zero.

The horizontal lift is given by the section of curve in base manifold times a group element. Why this is so needs investigation.

Curiosity

 

The important thing is not to stop questioning… Never lose a holy curiosity.
Albert Einstein

 

I used to be curious about things around me. I remember imagining and wondering about the universe and its state when I was 8. Small kids seem to be curious creature but their curiosity tend to disappear when they grow up. Why is this so? I reckon the education system is to be blamed for this. Of course mankind has not found the best way to educate children and maybe there is no best way to do this since every individual is unique. Put aside the question of how to educate the next generation properly, it might be more useful to consider how to develop our curiosity for those that have more or less lost it.

The website http://www.lifehack.org/articles/productivity/4-reasons-why-curiosity-is-important-and-how-to-develop-it.html has an article on how to be curious. The most important point perhaps, is to have an open mind. We do not and will not have answers to everything. We do have some theories(especially scientific ones) that, after checking with experience/experiments are considered facts. But we can never be certain about them since we might encounter some new phenomena which contradict or in conflict with our beliefs. This is interesting. We will then need to modify or even develop new explanations for these new phenomena.

 

 

Maximum efficiency engine

Just for the fun of it, let’s say I would like to build an engine. A good idea to obtain energy is to extract them from a heat source. Can I then extract certain amount of energy from the heat source and convert all of it into useful work? This will amount to 100% efficiency. I don’t think so. This I think is given by the 2nd law of thermodynamics, the Kelvin statement if I am not wrong.

If that’s so, can I build an engine that has maximum efficiency while obeying the 2nd law of thermodynamics? I think the search for this answer led Carnot to his Carnot engine. Maybe some assumptions were added to build this engine I will have to find out. I think one of the assumption is reversibility. Somehow this makes the engine efficient and no other types of engine can be more efficient than Carnot engine. This I am sure is the conclusion of Carnot’s theorem.

So, my plan is to find out under what conditions and assumptions can we produce an engine with maximum efficiency. Try to make sense of why these conditions or assumptions were chosen. Anything new we can learn from all these. Entropy somehow will appear and it plays a significant role in physics. Imagine that we have the concept of entropy arising from the practical issue of making an engine with maximum efficiency.

Some questions and ideas on Steifel manifold(I)

For Euclidean space, say , we can have vectors v such that |v|=1. These vectors form unit sphere . Transformation of these vectors can be obtained and is denoted by . Question arises. Why these unit vectors form unit sphere? Answer lies in the fact that we want to study about the properties of Stefiel manifold. And Steifel manifold is a manifold is a manifold that has something to do with vectors through the origin. I shall include what is defined as Steifel manifold here to be precise.

Steifel manifold V_k(Rⁿ) is the set of all orthonormal k-frames in  , where k-frame is basically k linearly independent basis vectors. Another question arises. Why do we concern about k dimensional instead of the full n dimensional basis vectors? This will have to be answered soon.

Reading: http://phys.org/news/2012-05-advances-mathematical-description-motion.html

Advances in mathematical description of motion

What is sympletic geometry? Why is it only used to describe even dimensional objects? The article suggested that length and angle are 1D objects (need to investigate this).

One interesting remark that caught my attention was that Lie group is used to investigate symmetries of geometrical structures. The article mentioned that the study of geometric space can be reduced to a smaller dimensional space. In the article also mentioned about cotangent bundles. My guess is that coset manifold is used to study the geometrical space. More importantly is how they extract useful and interesting informations by studying these reduced space. 

Reading: http://phys.org/news/2012-05-advances-mathematical-description-motion.html

Advances in mathematical description of motion

Complex mathematical investigation of problems relevant to classical and quantum mechanics by EU-funded researchers has led to insight regarding instabilities of dynamic systems. This is important for descriptions of various phenomena including planetary and stellar evolution.

Well known Euclidean geometry is used to measure one-dimensional (1D) quantities such as length or angle. Symplectic geometry is used to describe even-dimensional (such as 2D, 4D and 6D) objects.

The concept of symplectic structure arose from the study of classical mechanical systems such as planets orbiting the sun, oscillating pendulums and Newton’s falling apple. The trajectory of such systems is well defined if one knows two pieces of information, position and velocity (or, more precisely, momentum).

Quantum physics and Heisenberg’s Uncertainty Principle led to a modification of mathematics. Following the above example, a particle could no longer be considered as occupying a single point but rather as lying in a region of space, defined by two position coordinates and two velocity coordinates (four dimensions).

Continued evolution of mathematical theories led to the use of so-called Lie groups for the study of symmetries of geometric structures because it enabled ‘reducing’ the geometric space (dimension) under study to a much smaller one without sacrificing accuracy and clarity.

However, the observation of mathematical singularities, or points at which a given mathematical object is undefined or fails to be mathematically ‘well-behaved’, resulted in important effects on the dynamical stability of mechanical systems as well as instability of mathematical solutions.

The complex mathematics defining mechanical and dynamical systems was the focus of the ‘Hamiltonian actions and their singularities’ (Hamacsis) project, given that Hamiltonian equations provide a way of connecting classical mechanics with quantum mechanics.

Among the many insights gained by Hamacsis, extensive investigation of so-called cotangent bundles of vectors provided important descriptions of their reduced spaces in the cases of symmetric actions and singularities.

In addition, researchers explicitly showed how singularities of symmetric actions in several types of mechanical and dynamical systems affect stability properties. This is applicable to physically observed steady motions such as that in uniformly rotating bodies and planetary and stellar evolution.

The complex and innovative mathematics behind the Hamacsis project resulted in numerous publications in peer reviewed scientific journals. Outcomes significantly enhance our understanding of classical and quantum mechanics related to the motion of complex dynamical systems.

Some ideas on coset manifold (The use of Mathematica)

The computation of various mathematical objects when we are dealing with coset manifold can be tedious. For the case of Lie group G=SU(2), the matrices involved for example are still manageable. But this is not in general true. For instance, the Steifel manifold has 8 generators and dimension of matrices is higher than 2.

The plan is to compute the exponentiation of generators first. This can help us obtain the induced vector field. We could try to compute the derivatives necessary to obtain the induced vector fields using Mathematica. The exponentiation of matrix is easy to compute and it is given by the code: exponentiation of pauli matrix

Some ideas on coset manifold (Generators, integral curve and vector fields)

Consider a Lie group G=SU(2). This group has 3 generators namely the Pauli matrices. How do we know that they are the generators for this group? What is the use of generators? Apparently, they generate the group elements.

Knowing the generators allow us to evaluate orbits in the group. Also, from these orbits, we can obtain induced vectors at the identity of the group. What is the meaning of having orbits in a group?

I find it strange to perceive the idea of vectors and vector fields for Lie groups. But another thought sort of clear the obstacle for me. Euclidean space is a differential manifold. We can define vectors and vector fields on each point in the manifold. Since Lie group is also a differential manifold, we can define vectors and vector fields as well.

For example, the generator can be exponentiated to give the group element. The representation of group elements generated by   is given by

This gives us the integral curve or the orbit of the Lie group. Very simply, the vector at identity can be obtained once the integral curve is found.