Why do we use 360 degrees for a complete circle?

Sumerians, Greeks and Babylonians were the forerunners in observing the astronomical objects, may be for omens initially and later for calculating time. Sumerians observed that the sun takes a circular path and it takes approximately 360 days for one cycle (which is exactly 365.15 days which we call a year). This may the starting point for using 360. Obviously, others may have tried to change this to some other number. However, there are many reasons to stick to 360 degrees.

• It is divisible by 1, 2, 3, 4, 6, 8, 9, 10, 12, 16.
• Babylonians used base 60 numbers which is favourable to 360 degree
• The Greeks devided the the total number of days in to 24 hours.
• Later Babylonians devided each hour in to 60 minutes and each minute in to 60 seconds (which is their base system)

Vectors

What is a vector?

In everyday life, we measure things. For example, what is the size of something, what is the volume (how many liters) of the water bottle. When we buy things from shops, we ask 100 grams or 1 Kg etc. Assume that we need sugar. We will use just a number for example 2 Kg. But, there are some quantities which need not only a number but direction also. For example, if we ask where is this building, we may get a response like this: Go straight in this street. At the end, turn left and then first right. There you can find that building. Look at this explanation. Here, the location  (length of the distance is hidden) includes right and left. These are the directions.

This is in everyday life. In physics, there are so many such quantities which need direction as well as some number. These are called vectors. There are quantities for which just a number is enough to denote them. Those are called scalars.  Now what are the vector quantities?

• velocity
• momentum
• angular momentum
• acceleration
• force
• torque

Symmetry and Emmy Noether's theorem

Emmy Noether's theorems provide profound relation between a system having symmetry and constancy of a physical quantity. That is, there is a corresponding conserved quantity for any symmetry the system posses.

Following video from the YouTube channel "Looking Glass Universe". There, the title says "The most beautiful idea in physics - Noether's Theorem". The concept of symmetry is one of the ways to attack a problem as Feynman says.

Here is the video.

An easy way to find Latex code

There exists an excellent site on which you can find out the key for most of the $LaTeX$ comments.

The site is Detexify Latex handwriting.

The page says:

"Anyone who works with LaTeX knows how time-consuming it can be to find a symbol in symbols-a4.pdf that you just can't memorize. Detexify is an attempt to simplify this search."

What you have to do is just draw the symbol (on the portion given in the site) for which you want to know the comment.

You will be given the latex-code for that symbol. 

The page says "Philipp Kühl had the initial idea and Daniel Kirsch made it happen". Thanks to them.

Conservation of angular momentum

It is interesting to see the demo for "Conservation of angular momentum". The MIT2K series has an excellent video on the conservation of angular momentum.

 When he rotate the wheel, it has an angular momentum say $L$. The angular moment is given by $L=r\times \omega$ where $\omega$ is the angular frequency (which can be rewritten as $2\pi \nu$). This quantity is constant as for as no other forces acting on it. Now, when he flip the wheel, what happens? Isn't that really amazing? He starts to rotate. Why he starts to rotate? Because, when he rotate the wheel, the angular momentum directs downward (previously upward). Since angular momentum is conserved, it has to still $L$. So, the system make the demostrator to roate (with an angular momentum of $2L$ so that the total angular momentum is still $2L-L=L$. Gotcha...

Isn't that really amazing!!!


Questions:
How angular momentum conservation plays role in magnetism?
In a magnet, the angular momenta of all the electrons is alighted up. If we flip the magnet, does this has the same effect (i mean similar effect) as in the case of the rotating wheel here?

Is there any relation to the quantum hall effect, fractional quantum hall effect?

Notes on Fermi-Dirac distribution

As per my knowledge, in our universe, number of Fermions are more than Bosons (correct me if I am wrong). Apart from this, my interest to write on this topic is rooted on the fact that atoms consists of electrons, nucleons (proton and neutron), which in turn made of quarks which are also Fermions. Thus, in this page, I try to write on Fermions and their distribution function.

The Fermion distribution function at some temperature $T_{balance between the energy and entroy}$ is given by:

Notes on Dark matter and Dark energy

Dark matter and dark energy are one of the intriguing problems which keep awake theoretical physicists.

See this picture for a schematic of the accelerated expansion of the universe due to dark energy.

There are many attempts to demystify the truth behind these concepts. For first account on these topics, click following links from Wikipedia. I try to give a clear picture in the coming days.

1. Dark Matter
2. Dark Energy

There are attempts to unify these two concepts. For example, a recent arXiv pre-print titled "Unified description of dark energy and dark matter in mimetic matter model" by Jiro Matsumoto from Institute of Physics, Kazan Federal University, Russia attempts in this direction.

The abstract reads as follows:

"The existence of dark matter and dark energy in cosmology is implied by various observations, however, they are still unclear because they have not been directly detected. In this Letter, an unified model of dark energy and dark matter that can explain the evolution history of the Universe later than inflationary era, the time evolution of the growth rate function of the matter density contrast, the flat rotation curves of the spiral galaxies, and the gravitational experiments in the solar system is proposed in mimetic matter model."



While discussing dark matter and dark energy, we may need to consider the cosmic mirowave backround radiation (CMB radiation). In general, the CMB radiation picture is shown with a lot of variations. But, it is a difference with the actual image. The real image is completely red. What is shown in the Figure is just the variation (less than 4 K ?).

If dark matter is a boson, its masss  in lower limit is 10>-22 eV.
If it is Fermion, its Fermion, is lower limit is a few eVs.

The upper limit for the mass is 10^3 solar masses (huge!!!!).

Types of Dark Matter:

Bananas Dark Matter:



(This page will be updated gradually...)

Heizenberg's Uncertainty Principle

 Nobel laureate Weinberg (Ref. 7 of the English translation of the original paper by Heisenberg's paper)  has written as follows: ‘If the reader is mystified at what Heisenberg was doing, he or she is not alone. I have tried several times to read the paper that Heisenberg wrote on returning from Heligoland, and, although I think I understand quantum mechanics, I have never understood Heisenberg’s motivations for the mathematical steps in his paper. Theoretical physicists in their most successful work tend to play one of two roles: they are either sages or magicians....It is usually not difficult to understand the papers of sage-physicists, but the papers of magicianphysicists are often incomprehensible. In that sense, Heisenberg’s 1925 paper was pure magic."

So, it is very difficult to understand the Uncertainty principle from original paper. The paper in the above link try to give the method that Heisenberg may have used. 

Topology

Wikipedia article on topology  says:

• "Topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing."
• "Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space, dimension, and transformation"

Spend some solid amount of time to get a clear picture on Topology. 

Topology is a rich field and has many sub-fields. 

• General topology, also called point-set topology, establishes the foundational aspects of topology and investigates properties of topological spaces and concepts inherent to topological spaces. It defines the basic notions used in all other branches of topology (including concepts like compactness and connectedness).
• Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology and homotopy groups.
• Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.
• Geometric topology primarily studies manifolds and their embeddings (placements) in other manifolds. A particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots.

(This page need updates. Please come back later or go through Wikipedia article or other sources)

The pair of words that always confuse for many people

There are some words and phrases which is always confusing to us (or at least for me 😊).

Here is some of the list. It may take a while to understand the difference between them.

You can comment below to add more similar words.

• centripetal force and centrifugal force
• cpu and core
• affect and effect
• principal and principle
• whether and weather
• potential and potential energy

Links to Colloquium around the world

Colloquium is an excellent forum for updating and to get many ideas for solving problems and also to find out interdisciplinary problems. Almost all prominent universities conduct weekly colloquium. Here, I try to list them as I come across.

Following is the lists of colloquium around the world (Physics and Mathematics). I provide links for the university pages where videos/video-links are available.

• The Zurich Physics Colloquium (ETH Z$\ddot{u}$rich)
• Sommerfeld Theory Colloquium (Ludwig Maximilian University of Munich. A complete list of all the lectures, schools and workshop from Arnold Sommerfield center may be accessed from this page)

(This page will be updated whenever possible. Thanks for visiting this page.)

Linux tips basic comments (ls, mv, cp)

To rename a directory 

mv /home/user/oldname /home/user/newname

Formalism and Interpretation of Quantum Mechanics

We saw that there are many formalism for classical mechanics. Similarly, there are many formalism for quantum mechanics. Quantum mechanics is one of the successful theories that describe our nature. Albeit its success and many formalism, the interpretation of quantum mechanics is settled yet.

Different formalism of Quantum Mechanics are

1. Heisenberg 
2. Interaction 
3. Matrix  
4. Phase-space 
5. Schrödinger 
6. Sum-over-histories (path integral)
7. PT-symmetric Quanum Mechanics (Prof. Carl M Bender)
8. Ryu Sasaki formalism

The different interpretations of Quantum Mechanics are

1. Consistent histories 
2. Copenhagen interpretation
3. de Broglie–Bohm theory
4. Ensemble interpretation 
5. Hidden-variable theory 
6. Many-worlds interpretation
7. Objective collapse theory
8. Quantum Bayesianism
9. Quantum logic 
10. Relational quantum mechanics
11. Stochastic quantum mechanics
12. Scale relativity 
13. Transactional interpretation

Different formalism of Classical Mechanics

The study of mechanics which started with Galileo now rich in the sense that many different formalism are available. Here are the different formalism of Classical Mechanics.

• Newton's laws of motion
• Analytical mechanics
• Lagrangian mechanics
• Hamiltonian mechanics
• Routhian mechanics
• Hamilton–Jacobi equation
• Appell's equation of motion
• Udwadia–Kalaba equation
• Koopman–von Neumann mechanics

Here is the Wikipedia article to know more about Lagrangian mechanics. On the top right corner of the page, links are given for the different formalism (listed above)

(will be updated)

sudo apt auto-remove

What is the use of following command?

sudo apt auto-remove

This automatically removes unwanted packages from the linux distro.

Latex Tips (2)

How to add double dots on a character? For example, In the name Schrodinger, the correct writing would be a double dot on the letter "o". How this can be done?

To do this,

use \ddot{o}

This will give you ${\ddot{\text{o}}}$

In the name Schrodinger, use above syntax and you will get Schr${\ddot{\text{o}}}$dinger which you wanted.

Using Latex in blogspot

To add equations in the blogspot, there is a code provided by Prof. Matthew Leingang in Tex Stack Exchange site.

Click here to visit the Tex Stack Exchange page and answer.

Basically, what you need to do is

• Go to Theme
• Edit HTML
• Paste following code below <head> 
• Then save.

Now, you can use $LaTeX$ in posts.

<script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js">
MathJax.Hub.Config({
 extensions: ["tex2jax.js","TeX/AMSmath.js","TeX/AMSsymbols.js"],
 jax: ["input/TeX", "output/HTML-CSS"],
 tex2jax: {
     inlineMath: [ ['$','$'], ["\\(","\\)"] ],
     displayMath: [ ['$$','$$'], ["\\[","\\]"] ],
 },
 "HTML-CSS": { availableFonts: ["TeX"] }
});
</script>