calculus of variation – M Dash Foundation: C Cube Learning

May 29, 2015July 14, 2015

The method of Lagrange Multiplier ! Optimize multivariate functions.

Yesterday I learned the beautiful method of Lagrange Multiplier, to find the minima or maxima of multivariate functions. eg

If I ask you what’s the minimum value of a function, f(x,y) = xy, subject to the constraint 5x-y = 4, the answer would be -4/5. The method is this. Ensure that first-partials of function f(x,y) exist. That is fx = (∂f)/(∂x) is not zero, and fy = (∂f)/(∂y) is not zero. Else method won’t work.

For the given example the f(x,y) = xy, because fx = y and fy = x, our method will work.

According to the method, we have 3 equations:

(1) fx + λgx = 0

(2) fy + λgy = 0

(3) 5x − y = 4

December 16, 2013December 18, 2013

What are Gauge Potential, Whats a “Theory” in Physics?

Potential was, as said above, is, energy per unit mass or charge. So, we see that, in defining the higher quantity energy or potential (higher therefore closer to action, hence more fundamental or unified) we have to INTEGRATE the lower variable, here, Force, Field or (Any ) 3-vector. This entails therefore arbitrariness into the Physical solution when we solve for these quantities. These physical problems, as they involve differentials or integration, leads to a differential equation. Under further suitable physical conditions called eg laws of nature or physics, become whats called a wave-equation or for particles, equation of motion. We can say equation of motion for particles or equation of motion for waves if they are separate.

Now that we understand what are potential, field, vector and gradient and integral in relation to each other, comes requirements called as symmetry or laws of nature or laws of physics or in simple, boundary conditions to these differential equations known as, wave equation or equation of motion of particles OR waves. ( — which are separate so far )

These equations constrained by the conditions or restrictions which are attributes of physical observation, must therefore unite these variables (potential, field) into one entity which would satisfy the wave or particle equations of motion, the differential equations of motion in PARTICULAR ways only, known as Laws of Nature or Physics. So they become, from their 3-vector or scalar attributes, 4-vectors (or still higher, Tensors).

October 30, 2013October 30, 2013

Still some gravity !!

So talking about gravity fondly I have made a remark “Its not gravity that makes you fall”. You would be shocked as if you fell down but its not due to gravity but something else is at work, Am I a Physicist propounding a phantom theory? The actual statement I made is “Its not gravity that makes you fall, it only makes you fall FASTER”. That was known to Galileo, which we often very conveniently forget and make an erroneous statement that things fall because of Gravity. So it caters to the law of inertia also. So satellites would still fall irrespective of the absence or presence of Gravity. (although the inertia would mean they would be at absolute rest if we do not take into account enough of its past when something had hit it harder to slow it down or sped it up, in any case when enough history has been allowed to see that no forces or extra actions were disturbing its inertia, IT MUST BE MOVING AT UNIFORM MOTION without any debt to a force-bank called Gravity. If Gravity isn’t there the inertia is still the same and it would continue to be moving at the same speed, which as a specific possibility be ZERO)

So apart from the fact that Gravity only causes us to fall faster its also an erroneous fact to say: Gravity is caused by Masses. Or its innocuous looking twin-statement masses attract masses and thats called Gravity. It has two basic history why its often thought or said so. 1. History of how theory of Gravity developed. 2. History of how Gravity was taught.

Optical Path and Fermat’s Principle.

Mirror and Fermat’s principle: We can see ourselves in the mirror and take our mirror reflected selfie as a consequence of Fermat’s Principle, the topic of discussion of the blog.
…

Snell’s Law governs refraction which is adjustment of optical paths in in-homogeneous media because light can no more travel at its speed in free-space.

Snell’s Law and Refraction.
The above expression comes from Fermat’s optical theorem, called as “Fermat’s least time principle” which can in turn follow either from;

a. Huygens’s Principle; that light travels like spherical wave-fronts hence satisfies geometric rules or,

b. Principle of least or stationary action ( or Hamilton’s action principle ).

So in the beginning of our understanding we thought “Light travels a path which is shortest, or the least-path”. This is due to Heron of Alexandria, who lived between 10 AD and 70 AD. Then this path which is traversed by light was redefined to something called optical path, which led us to our understanding that “light travels the path of least time”.

All physical laws are derivable mathematically –with appropriate physical understanding, from Hamilton’s Action variation or Principle of Action, made to provide the least or stationary time. Then time is replaced by definition of action as the most general formulation of the law.

So all in all, path/distance >> Time >> Action . That is crudeness goes towards abstract physical understanding. This can be recognized as an attribute of unification and tells us why Mathematics reigns supreme in Physics.

In the last few weeks I am trying to understand why light traverses straight lines and why it refracts. The other day, I saw a little mug, floating inside a bucket full of water. Inside water any object would look shortened, this is known by a phenomenon called as refraction.

Lets envisage the phenomenon or observation, via this quickly reproducible trick. Lets dip our favorite pencil in a glass of water and another in a glass filled with air. This is how it looks, I quickly made this arrangement to which my 3 year old niece just glued with rapt attention.

October 3, 2013February 3, 2014

Equivalence from simple notions of Geometry ? Yes.

Now it may also be related that light bends in a denser media compared to a rarer media because an additional rotational force is working. In other words, the definition of straight line has to change in the medium, that is of different density, because path of light is changing. Light is the guy who suffers the least when something tries to buzz it, because its inertial property of mass is zero. The curvature of light or the bending or deflection known as Refraction is thus a measure of the sideways force or energy.

Thus speed of light in different media is a measure of this bending or curvature and is known as Snail’s law ( — Pun intended, its actually; Snell’s Law — ) . Automatically when distance and time have to readjust, to produce an angle or bending, known as refraction ( — possible because distance and time can produce an angle if they are equivalents — ) the speed must also change.

All these are inter related. Light refracts and its speed changes, in relative change in density of media because there are rotational or non-inertial effects.

( — or additional energy is available, or a force is acting to bring a curvature in light’s path, perhaps the electromagnetic effects of the molecules? Its not only distance or time that are equivalents, read one article of mine “All You Need To Know About Relativity” to understand; how energy and time and distance and in short all Physical Variables are equivalents of each other — )