# Posts from the ‘quantum mechanics’ Category

“Why are there so many crackpots when it comes to science or even anything intellectual? Because its difficult to be correct and easy to be wrong. Most people are wrong without knowing it because they chose the wrong. [in the context of Higgs Boson: eg If they had taken difficulties they would have become Peter Higgs by now. But they gave up, became incorrect and fizzled out.]”

an excerpt from a discussion on Facebook.

Manmohan Dash: Sheila Whitaker and Anonymous: Energy is not the most basic, it would be action which is the time integral of energy. Action is the total amount of force present in space-time. Energy is only the force present in space and not in time.

Branko Radoš: May be distortion of the geometry of empty space in the multi-dimensionality of our environment gives the effect of the existence of energy, particles and time.
Manmohan Dash‎ @Branko Rados; particles are ripples of field/force/energy and field/force/energy are functions of space and time and there are at-least 4 dimensions of space-time as we can verify now
Branko Radoš: Yes i know, but in lower scale, field, force, energy and time is distortion of multidimensional space.
Manmohan Dash: no you don’t know
Manmohan Dash: your both statements are flawed in the formal language of physics.
Manmohan Dash: “but in lower scale, field, force, energy and time is distortion of multidimensional space.” should be >> “but in lower scale, field, force, energy are distortion of multidimensional space and time
Manmohan Dash: here is the general idea in much simpler way
“Particles such as electrons are ripples or waves of field/force/energy and field/force/energy are functions of space, time and their derivatives and integrals. [in-fact force/field and energy are itself integrals over space or time of their derivatives] Basically differential equations are obtained because derivatives and integrations are involved and when the solution is obtained from a differential equation that is valid for both waves and particles, these solutions represent these particles which are also ripples or waves.”
Manmohan Dash: here is a general formalism of Physics [somewhat technical at times, but I guess you will get it ] http://mdashf.org/2012/07/01/why-is-energy-conserved/
Why is energy conserved.
mdashf.org
Branko Radoš: Sound interesting, but, hmm function of  the flow of time, seems the same as the result of multidimensional distortion in one way, but clearly not to say that there could be different.
Manmohan Dash: what I told you is the basic formalism of Newtonian Physics, Relativity Physics and Quantum Mechanics and there is only one way where you have “traversing space = distance“, “traversing/passage/flow of time = time/duration” their differentials or increments, their integrals = energy, force etc. When you have all these you have relations among these that are called equations of motion. You solve them for x, t etc and get e.g. energy in terms of x and t. So there is absolutely no confusion or different way here. Since energy is a function of space and time which are 4-dimensional the distortion you are talking about is the disturbance/ripple or wave. These are also particles because the solution satisfies Schrödinger equation. Hence ripples are of space time and higher order derivatives etc.
The comments below of anonymous depicts how careless and virulent the commenter is. I know nothing but shout at the top of my tone and want to supply idiotic and fact-less arguments against whats known and practised by the most formal frameworks of modern Physics. [Virulence: The ability of an agent of infection to produce disease]

The irreverent style in which the dots are sprayed just tells they are here for a reason, to gain some machismo by creating malicious environ. As I said virulent. But I will take some dots away so it at-least looks somewhat organized. They were typed randomly which is when you just want the others to believe you are some kind of great geek. Actually I got a message about this fellow who I will call Anonymous [I know his name, but what good] or Mr Fool as conscience permits me: this fellow is a crackpot and everyone knows this. But when you are running a good discourse on something such fellas just waiting at some corner to create virulent atmosphere.

Anonymous: Good morning everyone … What fun … Mr. Dash … If I may … I believe you have been spinning your wheels a  bit … The Higgs question … relates to mass … As to dimensions perhaps a mirror could … reflect … the true nature of how many there are or … are not … Mr. Dash … A very smart man once said … If you can’t explain it in a simple way … You don’t understand it … The language of Physics often throws many people off … When I was a young boy … Some friends and I created our own language … so only we would know what we were talking about … I could use it now if you like. I would hope we can keep it simple enough so everyone can play … and understand the debate / talk we’re having … Our … I could write a large book on how to cross the street if you like … And make it very … complicated … As for energy … you can’t have … ” action ” without it … Or for that matter .. mass … And Sheila Whitaker … Albert … who I love … Only got part of it right … String Theory … explains the rest … If I made any spelling mistakes please pardon me … I’m  working on my first cup of coffee still / energy … ~*

Mr Fool: Well … Mr. Dash … Don’t you have a pencil up your tight little ass … As with most arrogant fools like yourself … Your clueless about what I do our don’t know … As for what I do know about … Physics … I can see I’ve forgotten a lot more then you knowAnd that ant much … You sink to the classic form of an over inflated clown who is on the wrong end of what was a friendly debate until you came alone … You have the classic problem … You can’t  … ” PROVE ” … all your BSI can and have many time … With many more … “Educated ” … then you … You did do a great job of proving your … Just another arrogant asshole with a pencil in it … So bugger off dick head if you can’t be polite !

So the fool thinks whats the formalism of Physics written and proved million times in 1000s of text book I have to prove it again and its my BS. Anyway can’t dwell too much on fools. But just wanted to catch this amazing fool. Helps. [because we are not here promoting ourselves through some suave everyday conversation, as a physicist I and many around the world are working our arses off to bring the best that can be passed to the community/audience. ]

So upon such atrocious and virulent responses I thought I would use some of my trash-rhetorics. I wrote but stopped myself and did not post what I thought. But I see that I am a visionary. Several hours later I am writing this so you can read this. Not the first time or just in morning time when someone on Facebook used such malicious/abusive thoughts against me. Has happened 4 or 5 times at-least. When you say something that deeply throws off disagreement at someone’s strongly held prejudice they start such attacks. Man is a strange animal. One time someone started saying something abusive because some folks believe everything Ambedkar says is sacrosanct. Educated people don’t say much abusive language but they start to become really angry if something is said that does not cut with what they believe to be true: say it on Gandhi or even Budha or just about anybody, lets say Amir Khan, they will start throwing stones at you. Whats fucking wrong with you? Why are you wasting your time in protecting a dead soul? Couple times they started abusing when I tried to explain a physics concept or just wanted the other person to be a little clear. This is what you do: “yeah so you are saying “…”. Is that impolite language? The other person starts spraying his abusive bullets. But why? Anyway. Oh yeah one time a guy just came like that and started saying hipster words. Ofcourse they don’t realize one thing, their fun must not come from another person’s wallet. But then we can’t expect anything about the world can we. If any I would expect the world to be flat so I don’t have to spend energy.

SO here is what I would have told him except I didn’t but its funny so I am sharing with you.

“Hey son of a ** you think you are really good at that pencil up the ass joke? Here is some for ya. [and now I notice this guy’s English is so rotten, did he get that from a heap of used condoms?] You should have died within your mother’s womb. This is how. When you were being born and trying to take an exit to the outside world your mother was getting d** around in all holes. So you could not come out and died inside. asshole”

Some bonus: if you like to read.

Its not Physics Department’s problem if people don’t understand Maths. It becomes then everyone’s. [If it can come from Maths and rest on Physics won’t it wake up ever and go around the town? the problem? even to politics or vegetablenism?]

Everyone thinks Energy is the quantity that decides everything in Physics. But thats so not true. It would be action which is the time integral of energy. Action is the total amount of force present in space-time. Energy is only the total force present in space and not in time. Also Relativity Theory and Quantum Mechanics are based on formalism of Action and not Energy, in general. In particular… or less general cases of-course you can deal with the physical systems by Energy. In-fact someday we may need higher definitions such as integrals of action [which can now only be integrated over speed or acceleration] for better physical understanding. I call the integration wrt speed traction and integral wrt acceleration attraction. Anyway action and attraction are not today’s formalism and one needs serious theoretical developments. for this.
Modern Physics started as differential equation and serves as everyone’s pep talk. Even those who hardly understand a differentiation from integration. Which is fun. Because the world is saying you can’t keep us away from the fun just because its mathematics. Put some oil in the fire and make it even look like a God or something, but everyone loves to talk about something that everyone else is talking about. Lest we are all too lonely a planet.

[As an analogy] You have to think of Higgs as iron balls that can move on strings like beads. Whereever they go, the strings become heavier and the tiny little balls that can be there gain the weight of this iron ball. In other words only when the iron ball stretches the string with its heavy mass anything else also gain that heaviness. Higgs can give mass to other particles because its a massive particle moving through the “strings” [paths] of spacetime.

Higgs is said to be giving mass to other elementary particles in the world because of the scheme in which elementary particles share their energy and mass among each other. In other words every particle is giving mass to every other particle, but Higgs which is a special particle was missing so far. Most likely they have discovered it at CERN experiment. But every particle gives every other mass simply because they exert force on each other [or share energy with each other equivalently]

This confusion comes because mass of particles are a constant quantity [rest mass]. Hence if a particular “guy” is missing from a sports and you have planned everything wrt that guy having to deliver some important aspects then all the other aspects will be impacted. eg If Mr Rakesh didn’t give milk to everyone in town they will go hungry. But everyone might still get their milk through other possible ways. It would be misleading to say Higgs is the guy who gives mass to the particle in the sense that if it does not give mass the particles won’t get the mass. Higgs ALSO gives mass to other particles is the right way to say it. If Higgs won’t give mass the scheme would break and you have to look for fixing it. Much like if Rakesh is sick find another person or go without milk for sometime.  Neither is the Lagrangian represented as a Higgs Lagrangian through many images is any special. In-fact its a general Lagrangian not anything to do with just Higgs unless Higgs purported properties are also implemented.

And the mechanism of spontaneous symmetry breaking is properties of not just Higgs, its a general property. When the nature of a force changes the nature of underlying symmetry change, which through an association of equation of motion with various groups, such as SU(2) reflects this fact.

What is the meaning when we say we have found a new particle in particle physics? And what are new forces?

A new force is different from a new particle. A new force may be explained by older [that is known] particles as much as it can be by new particles if necessary. All we would see is a force which was hitherto unknown but its sources were known. Its possible that new forms of fundamental forces do exist, something I had “said” first time in september 2008. It would just mean a new source of energy. The mathematical form of energy dependence on location and the rate at which this location is changing etc is different which is why we would call it a different force than what we have known and hence a new force.

So forces are to be associated with Potential Energy only, and only then they will give us the knowledge: a new or known force. For that one subtracts the energy thats coming from pure motion of an object/system from the total energy. Once that is so, one could see if what we observe is explainable by all the terms the potential energy constitutes of. In other words eg if the potential energy that we see is fitting to the “mass” terms? There could be several mass terms and they each correspond to a particle among many.

Sometimes it happens that all the mass terms are fitted/satisfied when we try to distribute the energy to their sources and it happens that we know all the particles or mass-sources. But sometimes it happens that to satisfyingly explain something that we observe we do not see any mass/particle in our list of known particles which are literally 1000s in number .. [See Lund particle scheme] This is what data-analysis in particle physics is, you distribute the energy you observe in a process to various sources that you know exist. The sources that exist are literally 1000s.

So Physicists through years of data analysis iterate through all these particle’s mass and other properties and see if an energy content is fitting to the one that we would reconstruct from the different masses [or particles ] we know.

When we have run out of everything yet can’t explain some part of the energy we understand that some new particles are to be found which we do not know so far.

So Higgs particle is one such particle. If Higgs were not to be found all the theories we had constructed based on this which were used to explain other energy observations were going to fall apart. These theory and observation [and all the related tools, experiments, methods etc ] are together called Standard Model. And Standard Model would have fallen apart to some extent [not necessarily all: much in the same way your house does not break off because you had a gas-pipe blowing up somewhere, not necessarily]. When Higgs particle was found/confirmed, just today, and there may be more to know about it, such as are there more kinds and so on, it meant all the mass-terms we had for particles that we know or that can be hypothetically known from such are now valid, to the extent that we have based our theories or tools etc on Higgs. And it is said that Higgs is central to the theory/tool/fact-list called Standard Model. Now that we find this new particle and realize that it is that central particle called Higgs our methods have just become confident, far more than if we wouldn’t have found. But not finding it wouldn’t have shattered Physics ro Standard Model. [Standard Model is the detailed and fundamental essence of all of Physics so to say, if something would not work in condense matter physics it would definitely change our understanding about all of Physics and Standard Model, a standard model is a standard consensus, much like a pact that everyone has been expected and/or seen to be obeying.]

So you can find new forces, you can find new particles, you can find new channels or reaction [a particular way to spend some energy] and so on. They are all envisaged really well in Standard Model.

So a particle directly corresponds to a mass [rest mass] and a force directly corresponds to a potential energy. New would be something that we hadn’t known so far. But in Physics Force/Energy and mass are two different variables. [They are equivalents per advance understanding but not equal, you can sit in your office and time passes by which is equivalent to you are sitting in your car and time passes by and they are equivalent sans the speed at which the car is moving and sans the other forms of energy that were dilating/contracting the time but they are not equal. One makes you sit in your house. The other lets you enjoy a pub]

Now that Force corresponds to energy and you want to make it correspond to potential energy only you have to subtract the energy due to motion only and that will for a good deal involve various transformations etc because you want to be very careful about everything. But once you have known the potential energy you have observed is a new form or not it may or may not come from a new particle. So there are two: “possibility”. In general an old force and a new force.  And old particle or new particle. And then the possibilities for a set of observations are a combination of these two or four as you wish.

Why is energy conserved. Is there something called Principle of Attraction?

Why is energy conserved. Well everything is conserved. Are you not? Are you nuts? No I am not. What are you not, conserved or nuts. I don’t know, you con-fusing me. You called me a con. You confused me and called me a nuts. I didn’t confuse you. It was the conversation. Well why did you ask two questions when one hasn’t been answered yet. Well I didn’t ask two. You asked one of them. Which one? I don’t remember. Lets go back. I don’t wanna. If you don’t wanna I don’t wanna.

And this is where the band starts its music.

Is there something called Principle of Attraction? Lets review what we mean. I never heard that attraction factor in Physics. Nobody said it to me. So I would bet it does not exist. And I would call you a lunatic for imagining diamonds in air and proposing the world to harness it. Are you looking for funds for it? Well you know if we can find the diamonds then guess what we will pay back the funds and we will still have enough to cruise around the world.

Well here is what it is.

The Physics. The Physics inc. The Physics inc. defines its ways. But unlike Misissippi girl it changes its ways. Physics inc changes its ways but nobody gets it.

Here is the way that hasn’t been changed in a long time. You start with a simple object and note that such an object is defined for its motion by whats called its location in space = x or the increments in its location which is called an infinitesimal distance =dx, the instantaneous time at which its motion is referred, t or the increments in its time called dt.

Thats it. And we would like to know all the object does in terms of x, t. I would not like it if my dear people sit in a car and it vanished into thin air and never came back. I would like to keep track of it, the car because I know my people would still be in it. I would like to keep track of satellites and missiles and airplanes, I would like to know whats happening around me and why its happening. They all started with the quest to solve for the trajectory and then became more complicated as the complexity of these objects or systems grew. Collectively they satisfy greatly the quest we had set on, since millenia to understand whats all that goes on in our Universe, in our close vicinity and in situations far off from us, as far as the extraneous of the galaxy in which we live and more and more and more and deeper and deeper.

Then x, t are not sufficient to describe such situations. But since its all systematic we know all that has been defined, its not a party or Ramstein music band where you forgot what happened yesterday. Its hello Physics Inc. Pay attention.

Now as we defined dx and dt we also note that their ratios or as is called rate of x wrt t is called speed = v, written often formally as x with a dot on it. x-dot. Its the first order time derivative of x. v=dx/dt=x-dot. Then comes its derivative v-dot = dv/dt = a = d(dx/dt)/dt = acceleration. We also form two quantities: from v we form mv = p = momentum by multiplying the mass into the velocity/speed. From a we form the same way F=ma=force. But force is also defined to be the time-rate of momentum p, or in other words the ratio of the increments dp and dt. F=dp/dt. This is called Newton’s 2nd law F=ma =dp/dt. Its called a law but strictly its a mathematical law so far and not one which describes Universe’s phenomena so it cannot be called a physical law or principle as of yet. This point originally raised by Feynman as far as I know [If David Mormon comes he will stick his cotton buds and wipe this out and claim he said it, kami nahin he halkaton ki duniya me] It will be a physical law when Force is well defined and the differential equations as obtained from F=ma=dp/dt leads to a viable solution and explains some observed characteristics.

Since F=dp/dt you can also note that p = int[Fdt] and somewhere here impulse gets defined, perhaps impulse is Fdt, I was in highschool 20 years ago and refreshing this is hard job. Impulse tells you a rapidness of a force as the time could be defined very short to compare rapidness of different forces. A bomb will shatter you and obviously the gross impulse is 1000s of time more than a knock by your friend. Hence so much caution for bombs. And terrorists for this reason can be called impulsive.

Now that you have Force F and its a time-rate of change of momentum p [also p-dot, time derivatives are denoted by dots] we also have a quantity whose derivative is Force F. But this time its a derivative wrt space/location or x. So Force is called a space-gradient as its a space-derivative of a quantity called energy. So Energy E=int[Fdx] {for comparison: p = int[Fdt]} or you can differentiate and have F = dE/dx. Note that E can come in various forms: from its special form called Kinetic Energy T [or KE or K] or its general form called Potential Energy called U [sometimes V or PE]. Which is exactly why you can also say Relativity is categorized as special and general theory of Relativity [Special theory pertains to speed v only and hence T=mvv/2=pp/2m and general theory pertains to potential energy of all sorts not just gravity, gravity is a special case of General Theory: interestingly described in article written yesterday: What is Theory of Relativity] . When I said E can come in various forms I meant it can come as work/internal energy and so on. Work is W=-int[Fdx]=-U.

Also a very interesting you will note in the above if you are attentive that E=int[Fdx] and p = int[Fdt] evidently means E and p are equivalents in Theory of Relativity as x and t are. This means {x,t} and {p,E} are 4-vectors in Theory of Relativity, this means this integration itself transforms as a 4 vector equation in Lorentz Transformation. Recognize it from any text and let me know. [I haven’t ever studied many advanced text of Physics] So some hints of Relativity are already there in old classical mechanics although we may not have noticed if Einstein didn’t work everything out. [Read the couple of articles that tells you why much of Relativistic ideas were present in old classical mechanics and nothing new to Relativity of Einstein although their exact forms changed in whats now called Einstein’s Relativity Theory” loosely Relativity Theory ]

So to write in the line of defn: x, t, dx, dt, dx/dt=v, dv/dt=a, p=mv=int[Fdt], F=ma=-dU/dx, U=-int[Fdx], L=T-U, H=T+U

where U=U(x,t,v, a) and each quantity can depend on the other through relations or laws that are present in nature. Its just in this statement one can recognize there is some concepts of Relativity Theory or as I expounded in “Wjat is Theory of Relativity” equivalence theory: U=potential energy is equivalent to energy coming from x, t, v and a. In-fact potential energy comes from your location or configuration as functions of time hence x, t and it depend son the speed of your frame of reference v or in other words one needs to account for kinetic energy from the total energy H and get U or one can also have an equivalent form of energy coming from acceleration. If a system is accelerated there is some unaccounted energy compared to if the a=0=uniform velocity which is hoe everything has been formulated, so unaccounted changes were accommodated by defining dv/dt =a and we cut short this change by saying there is no da/dt. WHat if da/dt=0 or a=constant. Why are Forces always constant or are they? We have been solving Newton’s equation of motion [Not Newton’s Law: F=ma, thanks Feynman] by assuming Forces that are not changing in time. If they are its already included by integrating wrt time p=int[Fdt] and if its changing over space we are integrating over space [concept of force-field] E=int[Fdx] in other words we have defined a quantity called energy such that its sitting at the top of everything so far, x, t, v, p, a, F. Any change will be accommodated into energy. What about a change in speed v affecting E? What about a change in a affecting E? [1st derivative and 2nd derivative of space wrt time not completely accounting the Energy] Then Energy would not be conserved. The answer perhaps lies in two facts. one: we solve each equation of motion in a particular system know as many forces as we know and define in terms of the above kinematical and dynamical quantity and see if everything is fine. If not may be we are missing a force or not taking effects into account. If we can solve based on this formalism we have almost always gotten our best answers although there may be a few anomalies/loopholes/unsolved problems.

Summary of what I said: Before giving you the 2nd reason let me give you a few more quantities that are used when solving physical problems. Since U is a function of x, t, v, a etc and U is equivalent to energy which strictly depends on a [noninertialness/rotation/pseudo forces etc and T is a very special form of energy and also used to specify the frame of reference with the caution that in Relativity theory mass is a variable and the recognition that other variables described here may also be functions of such: x, t, v, a, … we have a formalism at place [Einstein just didn’t like leaving any variable to be assumed as constant and pulled out of equations but started differentiating them and applied a few of his insights and grossly enough changed the course of Physics] Add to that the following idea first. Like we defined Energy as a total force we can define action as a total energy, but this time integrate wrt time. Energy is a total force in space and action is a total energy in time. In other words sometimes we integrate wrt space and sometimes we differentiate wrt time.  So we have action is total force in space and time. In the formalism of Physics action is the prime-most variable. If you solve a differential equation by using Newton’s law scheme which is a specialized system specific approach you get solution to differential equation [called equation of motion] but the more powerful method would be the principle of Action because it does not leave quantities unattended.

Whats then the 2nd reason Energy would not be conserved? Perhaps the above, that some quantities may not only be left unattended by applying heuristic methods such as that of Newton’s 2nd law but by allowing the possibility that this Action Principle itself may also be leaving variables unattended. Who knows.

Here are the rest of the quantities that will perhaps make the formalism a little more comprehensive. There are fields defined from the forces by dividing the charges or masses [electric field E is electric force F / charge q] and [potential V is potential energy U / mass m]. Gravity is just a special example of a potential energy and we have gravitational force, potential, potential energy and so on. Some more quantities are thusly E, B, Phi and A. Actually B is a field like E field [or its corresponding force or potential or potential energy etc are invoked], Phi and A come as an alternative formulation of E and B, called scalar and vector potential respectively. So these are potentials  which are either scalar or vector quantities and by treating them as potentials you can from our ensuing formalism get the E and B. [Potential is Energy by charge or mass and then differentiate to get force and divide again by charge or mass to get field B, E] But the advantage is wjat is called Gauge formalism symmetry. In this way a variety of Phi and A will produce the same E and B. But since changing Phi and A did not change E and B but made the purported solutions much easier to deal with this is called a gauge symmetry [an underlying quality or quantity remaining the same even if you submit them to some kind of transformations or change, this case: Phi and A changed, underlying quantity E and B did not, so Gauge transformation and gauge symmetry, Gauge is the name of a lady in the cyber world and physics has got nothing to do with it.] But note that Phi and A still conform to our hello physics inc. formalism. They are potentials. Also they satisfy whats called a wave equation. From the concepts of a wave you can define wavelength: lambda, wave number: k, frequency: neu/f and time period T. You can have a phase an amplitude which is the maximum separation of the wave from its mean position. Probably you do not have any more quantity in the formalism. Well Probability but that comes when you allow the wave to be a complex number function.

So by applying least variations on action S one gets a bunch of differential equations [for various systems] and solves them. These equations are called equations of motion. If they are defined for an object with localized attributes its called an equation of motion. If the object or system is not localized but have extended attributes then the differential equation is called wave equation of motion, but still comes from action principle. If the object is a dual-attribute: both localized and extended, obviously not necessarily both at the exact same instant of time, called the complementarity principle of Quantum Mechanics, obviously as a name sake because wave compliments a particle and vice versa but they do not replace each other or appear at the same time because that will make them the same object either localized or extended and one would lose the dual-attribute then the action principle gives you whats called a wave-particle equation of motion, Schrödinger’s equation or wave equation of motion in quantum mechanics.

Interestingly enough as I had noted recently [and dwelled upon it for sometime as I had asked such a question several years ago] why the Schrödinger’s equation of motion is a 1st order time derivative but a 2nd order time derivative in space? Where as the wave equation of motion of classical mechanics is a 2nd order time and 2nd order space differential equation? Thats because the particle equation of motion or simply the equation of motion in classical mechanics is a 1st order time derivative [along with 2nd orders as well of time] equation. So a wave-particle equation has to be 1st order time derivative: at-least that a good hint.

Here are their forms:

Particle: [1st order differentials in time]


Wave: [only 2nd order in space and time]



Wave-particle: [again 1st order in time, oops … there must be a particle here, 2nd order in space? ok there is a wave here too ]


Ok the list of quantities is not yet finished. You gonna have angular momentum and this is listed into two types: spin and orbital. And Energy, ang. momentum, p etc are sometimes called constants of motion. In some motions p is conserved, in some L=S+J is conserved. [L is not lagrangian = T+U and S is not action but spin, J is orbital, one can use small s and j and l to avoid this ambiguity !!]. But it is said that E is always conserved which is what this article wanted to explore.

I am thinking any more quantity? …

Ok perhaps not. The quantities that are conserved is connected to Noether’s theorem: for every symmetry transformation there is a corresponding quantity that is conserved and for every quantity thats conserved there is a symmetry that corresponds to this, OR: symmetry transformations and conservation are necessary and sufficient towards each other. So p, l and E etc are associated with corresponding transformations. eg p is conserved if translational symmetry is satisfied which means F=0, so dp/dt = 0 or p=constant. whats the big deal, nothing, its just a simpler form that we always studied. Similarly l = constant if torque = rXF is zero or there is rotational symmetry [rotational forces are zero]

Energy is conserved if there is time-translational symmetry. A space flipping is called  a parity symmetry, a space translation that leads to constant p is x’ =x + dx, a space rotation is theta’=theta+d[theta] leading to constant l. A time translation: t’=t+dt; leads to energy conservation or constat energy. But one shall recall that S=int[Edt] where E appears as total energy H or “differencive” energy  L. [absence of T or U make it a special case so E=T say]. Since S is submitted to least variation under which it must remain “minimum and constant over various paths”any difference of time must not change E. So “energy conservation” is embedded into the “principle of action” or the latter would not be valid. They are perhaps necessary and sufficient for a good formalism to work.

But this is good enough to see why time translation was inherently connected to Energy conservation. I have actually used least action principle to derive equivalence principle of Einstein [perhaps in a round about way which led to a simple insight that given to the constant of integration T and U must be equivalents or in other words there is no difference in their nature hence equivalence of T and U and hence equivalence of say m_inertial and m_gravity] As I have noted here and elsewhere the gravity is just a special case of U and it derives from acceleration=a in many forms such as rotation or pseudo forces and there is nothing in action principle that suggests they must be different. That is acceleration being a form of energy and pseudo acceleration being another they serve just like Gravity. There is nothing Einsteinian about Gravity or equivalence principle [and even time dilation itself] Its energy that warps space, time, speed, acceleration etc and warping means not having a flattened appearance or attribute or quantity, this is the basis of ordinary motion. Just from this you can build theory of relativity in alternative ways.

[In-fact I have obtained time dilation from a simple gravity force without using Relativity, I will post the calculations soon, but the ideas are described in article: http://mdashf.org/2012/06/27/time-dilation-all-energy-slows-down-clocks/ ]

Now that we see that energy conservation is evidently [manifestly] accommodated by the principle of stationary action we ask why Energy is the highest level of a physical quantity that must be conserved. eg why not we stopped at p or l or even v. Because we say these change when there are higher order changes present out of the system or unaccounted. So we went on integrating to find the higher order variable and currently we have Energy. But to note we integrated wrt either x or t, in steps. [Action is total force]

We never integrated wrt say v or a.

What if this way we account only for the action S thats fitting into our scheme. Perhaps nature has more and thats why our formalism are not the most consistent.

Lets define traction. Traction Z = int[Sdv] we integrate wrt v so S = dZ/dv, being a velocity derivative we called it traction.

Then we can define Attraction A = int[Zda], this accommodates the changing accelerations. Z = dA/da.  Attraction is the total force if you consider x,t,v and a not just x,t. Then you can apply least attraction. A least attraction is what we are seeking in nature. We want to take that path for which attraction is the least and stationary over all available paths.

This formalism was neither thought nor explored ever. Perhaps this will lead to more physics insights unless a mathematician proves us wrong by saying why according to mathematical concepts of now we do not need so many variables. Perhaps we need these variables and mathematics and with it some physics will be changed.

Thats all I wanted to say this after-noon about conservation of energy.

Gangsta rappin .. we started rappin.