June 30, 2018 Saturday

Bedtime Story 

Foundations of Lagrangian Mechanics

Tonight we shall continue with the last four reformulations of Mach’s principles.

Statement 8 – If you take away all matter, there is no space.

Statement 9 – Ω = 4ϖρGT² is a definite number, of order unity, where ρ is the mean density of matter in the universe, and t is the Hubble time (which is inverse of Hubble’s constant and rounds up to 14.4 billion years)

Statement 10 – The theory contains no absolute elements

Statement 11 – Overall rigid rotations and translations of a system are unobservable

With this I shall end my digression on Ernst Mach and return back to Lagrangian mechanics.            

Now we have enough background material to discuss the foundations of Lagrangian mechanics.

Any moving mechanical object, for instance, a swinging pendulum that has always been the subject of study for the classical physicists (and a favorite of quite a many examiners in many competitive entrance exams world over), can be tracked very accurately using Newtonian mechanics.

Newtonian solution of moving objects involves time-varying constraint forces that keep the body in the constrained motion.

Such forces include reaction force exerted by the wire on the pendulum or it could be the tension on the swinging rod.

Lagrangian mechanics does away with tackling forces on such moving objects.

It considers the path the particle (bodies are generally assumed as particles for simplifications) can take and then picks out some independent generalized coordinates that would suitably characterize its motion.

Generalized coordinates are not exactly the type of coordinates that you study in Cartesian geometry; they allude to certain parameters that describe the configuration of the system with respect to some reference configuration.

These parameters satisfy mathematical constraints such that they represent a specific manifold/space.

This may be confusing to you very understandably since to most of us Euclidian space is how we perceive the world.

Even in our daily lives knowing that the Earth is a sphere, for all practical purposes on our small local scale, it is flat.

This makes our planet Earth a topological manifold which can be generalized to be flat locally in spite of being a sphere factually.

Manifold is a mathematical equivalent of the physical space but it is a more general term implying topological space.

The concept of manifold allows our mind to look into space in a way that is not intuitive to our evolved minds since for its survival and procreation it never had to consider any more dimensions than three.

To explain this idea in a more visual way, we will need to consider a big thick book of the beautiful Bartholomew Atlas and a globe of the earth often seen in many plush offices often accurately depicted with the axial tilt.

I shall proceed to give you a fairly-intuitive satisfactory understanding of the concept of the manifold in the nights to come.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night Mon Ami and my fellow cousin ape.

                           

  

                

             

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Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

June 29, 2018 Friday

Bedtime Story 

Pondering over Mach's Cryptic Verse

Strange enough, even if you look very attentively at the words of Ernst Mach, they do not seem to be asking anything wise or profound.

Please read those lines of Mach once again, pause and give it some thought.

“Investigator must feel the need of…knowledge of the immediate connections, say, of the masses of the universe.

There will hover before him as an ideal insight into the principles of the whole matter.

From which accelerated and inertial motions will result in the same way.”

There is some kind very vague reference of ‘masses of the universe’ (which itself sounds very ambiguous) with the Newton’s concept of inertia related to inertia of accelerating bodies on planet Earth.

And yet, it is exactly such an ill-defined statement of Ernst Mach, lacking both in mathematical rigor or logical depth, that Einstein derived his explosive idea or at least, this is who the modest man gives credit to.

Mach’s principle in modern physics can be stated in several ways far better then he himself came up with.

I found the following two general statements to be very decent generalizations of the Mach’s principle:

(a) Local physical laws are determined by the large-scale structure of the universe

(b) Mass out there influences inertia here.

Yet in spite of Mach’s ideas, the origin of inertia till date remains the most obscure subject in the theory of particles and fields.

Mach’s principle though it gives a hint of the origins of inertia, does not still fully explain its origins.

There are at least eleven ways in which Mach’s Principle can be rephrased and restated.

It is an interesting exercise to go through each of them to see how so disparately looking statements are in truth reflections of the same idea.

Statement 1 – The universe, as represented by the average motion of distant galaxies, does not appear to rotate relative to local inertial frames.

Statement 2 – Newton’s gravitation constant G is a dynamic field.

Statement 3- An isolated body in otherwise empty space has no inertia.

Statement 4 – Local inertial frames are affected by the cosmic motion and distribution of matter.

Statement 5 – The universe is spatially closed.

Statement 6 – The total energy, angular and linear momentum of the universe are zero.

Statement 7 – Inertial mass is affected by global distribution of matter.

We shall consider the next four statements of Mach’s principles in the nights to come.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night Mon Ami and my fellow cousin ape.

                           

  

                

             

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Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

June 28, 2018 Thursday

Bedtime Story 

The Question of the Origins of Inertia

Last night I had left you with Mach’s own statement that now goes as Mach’s principle which no doubt is very vague and hard to comprehend.

It obviously would not have made much sense to you unless you know the context in with it was made.

Even with the context known, it has been stated so vaguely that it can be interpreted in several ways.

We are talking about the mid or late nineteenth century when physicists and philosophers alike were feeling that there is something amiss about Newton’s explanation of the workings of universe.

There were many questions, even very fundamental concepts that were left both unanswered and unexplained.

One of such concepts among many others was the vexing subject of undefined inertia.

Inertia in classical mechanics is seen as the primary manifestation of mass which in turn is a quantitative property of any physical system that is measurable.

Just have a look at how Newton defined inertia in his classic text ‘Mathematical Principles of Natural Philosophy’ (1687):

“The vis insita, or innate force of matter, is a power of resisting by which every body, as much as in it lies, endeavors to preserve its present state, whether it be of rest or of moving uniformly forward in a straight line.”

Now, whenever you come across the use of the word ‘innate’, get cautious because it is a warning beacon that something has been left unexplained.

It only suggests that it is a subject open for investigation by fertile young minds.   

Where does inertia come from?

What is the meaning of absolute motion when it is clear that if there were to be a single moving particle in an empty universe the notion of motion becomes bereft of any sensible meaning? 

Just look at this apparently silly questions posed by Steven Weinberg in his 1972 book ‘Gravitation and Cosmology: Principles and Applications of the General theory of Relativity’”

“You are standing in a field looking at the stars.

Your arms are resting freely at your side, and you see that the distant stars are not moving.

Now start spinning.

The stars are whirling around you and your arms are pulled away from your body.

Why should your arms be pulled away when the stars are whirling?

Why should they be dangling freely when the stars don’t move?”

These lines if asked by a child would instantly be dismissed as rubbish by any adult.

We shall continue to look into the Mach’s principle in the nights to come.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night Mon Ami and my fellow cousin ape.

                           

  

                

             

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Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

June 27, 2018 Wednesday

Bedtime Story 

Ernst Mach on Lagrange and Mécanique Analytique

We shall continue with what Lagrange had to say of the revolutionary nature of his own work that he published as Mécanique analytique:

“The methods that I explain require neither geometrical (he surely wasn’t very fond of geometry if you may recall), nor mechanical, constructions or reasoning, but only algebraical operations in accordance with regular and uniform procedure.

Those who love Analysis (he surely was deeply in love with it) will see with pleasure that Mechanics has become a branch of it, and will be grateful to me for having thus extended its domain.”

This complete absence of any form of modesty has a very sound reason; for the claim made by Lagrange is absolutely accurate.

Lagrange was essentially a very modest and shy man, but it is specially such unassuming men who are keenly aware of any kind of monumental intellectual work, either somebody else’s or their own.

If you see from his perspective, it was an achievement very close to his tastes; Here was a man who from his very childhood was fascinated with analysis and had an odd dislike bordering on contempt for geometry.

On these very lines Lagrange redefined the mechanics of Newton that to a large extent uses geometry in the form of vectors and trigonometry.

This is what Ernst Mach (a critique of Newtonian physics) had to say of Lagrangian mechanics.

“The value of his work [Mécanique Analytique] consists in the exposition of a general method by which every mechanical question may be stated in a single algebraic equation.

The entire history of any mechanical system, as for example, the solar system, may thus be condensed into a single sentence; and its detailed interpretation becomes simply a question of algebra.

No one who has not tried to cope with the difficulties presented by almost any mechanical problem can form a just appreciation of the great utility of such a labor-saving and thought-saving device.

It has been well called ‘a stupendous contribution to the economy of thought’.”

You might not be aware, but Ernst Mach is known for his Mach’s principle or Mach’s conjecture that was labeled by Einstein as such.

This principle became the guiding force for Einstein in framing his theory of relativity.

This is how Mach himself thought and wrote about it:

“Investigator must feel the need of…knowledge of the immediate connections, say, of the masses of the universe.

There will hover before him as an ideal insight into the principles of the whole matter. From which accelerated and inertial motions will result in the same way.”

I will try to explain to you in the nights to come what Mach was trying to convey.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night Mon Ami and my fellow cousin ape.

                           

  

                

             

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Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

June 26, 2018 Tuesday

Bedtime Story 

How Lagrange Related Himself to Newton

Last night we got a fair understanding of how analytical mechanics (both Lagrangian and Hamiltonian) differs from Newtonian mechanics.

No new views of classical physics need to be applied while applying these two other forms of mechanical analyses which means to say that the essential concepts of physics remains unchanged.

The best way to describe analytical mathematics of Lagrange and Hamilton is that they are a set of equivalent formalisms that have broad applications to classical mechanics.

The beauty is that this technique goes even further and the same principles are applicable to relativistic mechanics and general relativity.

With some alterations, these are even applicable to quantum mechanics and quantum field theory.

That is the power of mathematics for you.  

Many of you may feel that Lagrangian mechanics is merely a reformulation of Newtonian mechanics, which is not completely wrong but it goes further.

Lagrangian mechanics plays a vital role in providing a deeper understanding of physics.

It seems from one of the quotations of Lagrange that intellectually he considered himself at par to Newton.

So much so that he believed in his mind that he would have himself discovered what Newton did had Newton been not the first one to beat him to it.

This is not very overtly stated but the above implication is hard to miss for anyone who is skilled at reading between the lines. 

“Newton was the greatest genius that ever existed, and the most fortunate, for we cannot find more than once a system of the world to establish.”

Of course, Lagrange never lived to see the revolution of relativistic physics that would start some eight decades or so after his death.

He was right is saying that Newton was the greatest genius that ever existed, though we was surely wrong in his second half of the comment, as there indeed was someone who did found out more than once a system of the world that still boggles our minds.

Lagrange though was very confident of the revolutionary nature of his work in Mécanique analytique and this is what he had to say about it:

“We have already various treatises on Mechanics, but the plan of this one is entirely new.

I intend to reduce the theory of this Science, and the art of solving problems relating to it, to general formulae, the simple development of which provides all the equations necessary for the solution of each problem.

I hope that the manner in which I have tried to attain this object will leave nothing to be desired.

No diagrams will be found in this work.”

We shall continue with Lagrange’s assessment of his own work of analytical mechanics in the nights to come.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night Mon Ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

June 25, 2018 Monday

Bedtime Story 

Exploiting the Degrees of Freedom

Last night we were talking about the degrees of freedom a point particle would have in a three-dimensional Euclidean space.

At the minimum we will need to specify three numbers that would attest to its translational position.

Besides this, the particle can also undergo rotation and vibration operation or transformation. 

A vibration of a point particle at room temperature is miniscule and hence negligible and hence its degree of freedom with respect to vibration is zero.

It gains significance in thermodynamics as the temperature is one of principle quantities in the study of thermodynamics.

In statistical mechanics temperature is a measure of the mean kinetic energy of the classical fundamental particles of the system.

Hence while at higher temperatures the degree of freedom from vibration becomes critical, at room temperatures it can be ignored and can be assigned the value of zero.

Similarly, a particle as understood in the classical mechanics, will remain unchanged if it undergoes rotation and hence the degree of freedom due to rotation can also be assigned zero to a single point particle in space.

So the total degree of freedom of a point particle in a three-dimensional space adds up to 3 + 0 + 0 = 3.

Now for a moving particle in a three-dimensional space at a certain velocity the calculation of degrees of freedom gets trickier.

We know that a point particle would require three position co-ordinates to define its position.

Now it also has a certain velocity with speed and direction which has three components too.

This is a system that in respect to evolution of time is deterministic; meaning that if you know its location and velocity at one point then from it you can either go backward and know its past position or go forward and determine its future position.

Such a system will have six degrees of freedom; three from our previous calculation and three from the vector component at each point.

The analytical mechanics makes use of this degree of freedom concept as each system has constrains that limits its degree of freedom and this helps to reduce the number of coordinates to solve the equation of the system.

Analytical mechanics has been broadly divided into Lagrangian mechanics that is the subject of out bedtime story and Hamiltonian mechanics.

Lagrangian mechanics is that mechanics that rests on generalized coordinates and corresponding generalized velocities in configuration space.

Hamiltonian mechanics, on the other hand, rests on coordinates and corresponding momenta in phase space.

Both formulations are equivalent to each other and with Newtonian mechanics and hence contain the same information and yield the same solutions while working out the dynamics of a classical system.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night Mon Ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

June 24, 2018 Sunday

Bedtime Story 

Holonomic and Non-holonomic Constraints

For our story, it is important to know that constraint can either be holonomic or non-holonomic.

Without getting too technical, let me briefly state what is meant by holonomic constraint.

Once again, mathematics has to be introduced to define the idea of holonomic constraint.

A holonomic constraint is such that it satisfies the following function:

 f(x₁, x₂, x₃,…,x_N, t) = 0

This is to say that a holonomic constraint depends only on the coordinates x and time t.

What makes it special is that such a constraint does not depend on any derivative of time t such as velocity and higher derivatives such as acceleration.

This can be interpreted as a holonomic physical system being such whose constraints are only a function of scalars and not vectors.

On the other hand, a non-holonomic is a system whose constraints cannot be expressed in the above manner.

This is so because the final state of the system depends on just the coordinates but the intermediate values of its trajectory through the space. 

    

Another way to understand the idea of constraints in a mechanical system is through the concept of degrees of freedom.

Any physical system is characterized by degree of freedom that is an independent physical parameter which can formally describe a physical system.

One can also understand the degree of freedom as the minimum number of coordinates required to specify a configuration.

A single particle on a flat plane with only x and y axis would have two   degrees of freedom since just two coordinates are needed to specify its configuration.

Similarly, a single point particle in a three-dimensional space would have three degrees of freedom since three coordinates are needed to specify its configuration.

It can be any useful property of the system that is independent of variables.

To elaborate a little bit on this subject of degree of freedom, consider a gas particle.

The center of this particle has three degrees of freedom as a location of a point-article requires three position coordinates.

This would be its translational and a geometrical representation of this point.

The degree of freedom in a three dimensional pace can be further decomposed to rotation and vibrations.

A point particle will remain unchanged if it rotates and so its degree of freedom with relation to rotation is zero.

We shall continue out story on the degrees of freedom in the nights to come.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night Mon Ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

June 23, 2018 Saturday

Bedtime Story 

Constraints of a Physical System

One of the well known scalar quantities is the temperature since it just needs a single set of integers (along with the appropriate units) to quantify it.

Other such examples of scalars are well known to you and they include mass, charge, volume, time, speed and charge density.

Here a word of caution has to be made; the quantities that are scalars in classical mechanics have to undergo modifications in relativistic physics.

These classical scalar quantities will need to be combined with other quantities and then treated as 4-vectors or tensors in relativistic physics.

An example of this would be charge density at a point in a medium which is a total scalar in classical mechanics.

Yet in relativistic physics it needs to be combined with local current density which is a 3-vector to give a relativistic 4-vector.

Let us not go too much into relativistic physics as that will certainly confuse you (and me as well).

The point that I was making earlier was that in contrast to Newton, Lagrange developed mathematics in classical mechanics that would make use of scalar quantities to arrive at the same elegant solutions that Newton achieved.

The mathematics that was developed by Lagrange, Hamilton and others goes broadly by the name of analytical mechanics or sometimes more specifically as Lagrangian mechanics as homage to the founding father of this field of classical mathematical physics.  

While Newtonian mechanics attacks the problem of moving objects directly, Lagrangian mechanics does it more subtly making use of the constraints that are inherent in the physical systems.

What do we mean by constraints and how are they inherent in a physical system?

These are critical questions that every child should be made to understand in order to emphasize that nature operates on certain principles that cannot be defied and perhaps this can help take out magical thinking that most of come endowed with and then gets culturally reinforced and later permanently embedded mentally.

A constraint of any physical system in classical mechanics is any parameter that a system must obey.

Just to give one very banal example that you have often dreadfully encountered in your high school exams of a block of wood sliding down a slope marked across with several arrows pointing down, up and at right angle to the slope along with trigonometric functions attached to them with angle dependent on the angle of the slope.

In such a system what is the constraint?

The constraint is, particularly when it comes to deriving your solutions, is that the block of wood has to be in contact with the slope all the time.

The alternatives of the block flying around it or tunneling through it are not permissible.

          

Now this is a very loose definition of a constraint but you get the idea what is implied by a physical system having a complaint.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night Mon Ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg