Why is energy conserved? 5

Why is Energy Conserved?  — July 01, 2012.

Manmohan Dash,

Energy [or more general; action] distributes itself, through equation of motion, which is a differential equation obtained by applying principle of stationary action, onto, Hamiltonian Action Integral.

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 Mississippi 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. We, start with a simple object, and we 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 the millennium and more, 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 ratio, or as-is-called, rate of x wrt t, called speed = v, written often formally, as, x with a dot on it. xdot. Its the first order time derivative of x = v = dx dt = xdot. Then comes its derivative vdot = dv dt = a = d(dx dt)/dt = acceleration.

We also form, two quantities: from v we form m.v = p = momenta; by multiplying the mass into the velocity or speed. From a, we form the same way; F = m.a = 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 = m.a = 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 was originally raised by Feynman; as far as I know.

It (F = m.a = dp/dtwill be a physical law, when, Force is well defined (eg in terms of its dependence on space and time and their higher increments), and, the differential equations as obtained from F = m.a = dp/dt leads to, a viable solution and; explains some observed characteristics.

Since, F = dp/dt, you can also note that p = int [F.dt] and, somewhere here impulse gets defined, perhaps impulse is F.dt, I was in high-school, 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 alone, can be called impulsive.

Now that we have Force F, and its a time-rate-of-change-of-momentum p [also pdot, time derivatives are denoted by dots] we, also have; a quantity whose derivative is Force F. But, this time, its a derivative w.r.t. space or location x. So, Force is called a space-gradient; as, its a space-derivative of a quantity, called energy. So, Energy = int [F.dx] {for comparison: p = int [F.dt] } 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 K.E. or K] or its general form called Potential Energy called U [sometimes also, or P.E.]. 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, hence, T = m.v.v/2 = p.p/2.m and, General Theory pertains to potential energy of all sorts, not just gravitygravity is a special case of General Theory, interestingly enough, described in article written yesterday; What is Theory of Relativity.

When I said E can come in various form; I meant it can come as work, internal energy and so on. Work is W = – int [F.dx] = – U.

Also, its very interesting, you will note in the above, if you are attentive, that, E = int [F.dx] and p = int [F.dt] 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, under Lorentz Transformation. Recognize this, 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 might not have noticed; if Einstein didn’t work everything out. ( — In addition you may want to read, the couple of articles, recently written by me, that tells why much of Relativistic Ideas were present in Old Classical Mechanics, and this is nothing new to Relativity of Einstein although, their exact forms changed, in whats now called; Einstein’s Relativity Theory or loosely Relativity Theory )

So to write in the line of definition:

space and time points, and their differentials; x, t, dx, dt, dx/dt = v,

differentials of speed, acceleration; v, dv/dt = a,

momentum (from “lower” speed and “higher” force); p = m.v = int [F.dt],

force (from “lower” acceleration and “higher” energy-work); F = m.a = – dU/dx,

energy-work (from “lower” force)= – int [F.dx],

energy (in two useful forms) L = TU, H = T+U

Action (gets defined from “lower” energy; in two forms, as an integral) The End of Physics Formalism.

Lower: differential, Higher: Integral.

where = U (x, t, v, a) and each quantity, can depend on the other, through relations or laws that are valid in nature. Its just in this statement one can recognize, there is some concepts of Relativity Theory or as I expounded in “What is Theory of Relativity“, equivalence theory; = potential energy is equivalent to energy, coming from x, t, v and a.

In-fact potential energy comes from our location or configuration, as functions of time, hence x, t and it depends, on the speed of our 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 how, 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, = constant. Why; are Forces always constant or are they?

We have been solving Newton’s Equation of Motion (not Newton’s Law: F = m.a, thanks Feynman, explained above) by; assuming Forces that are not changing in time. If they are, such changes are already included, by integrating w.r.t. time, p = int [F.dt] and, if its changing over space, we are integrating over space, ( — concept of force-field) = int [F.dx] 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 for 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, kinematic 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 or 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 or 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 in 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. So instead of assuming that mass is a constant, as has been assumed for centuries preceding him, he considered it a variable and differentiated it, using a product rule and showed that, with his insights such as speed-of-light is a constant in wave theory of light; mass must depend on speed of objects. This is also a cursor to E = mcsquared, as; given a speed we have an energy. Basically he segregated the equivalent relations by being careful enough.

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 w.r.t. timeEnergy is a total force in space and action is a total energy in time. In other words, sometimes we integrate w.r.t. space and sometimes we differentiate w.r.t. time.  So we have; action is total force in space and time.

In the formalism of Physics, action is the prime-most variable. If we solve a differential equation, by using Newton’s law scheme, which is a specialized system specific approach we get solutions to differential equation ( — called as “equation of motion”) but; the more powerful method would be, the Principle of Action, because, it does not leave quantities unattended. (Action is then like Einstein)

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 (Phi, A) are potentials, which are either scalar or vector quantities, and by treating them as potentials; we 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 what 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.

Symmetry: an underlying quality or quantity remaining the same, even if we submit them to some kind of transformations or change. In this case: Phi and A changed, underlying quantity E and B did not, so Gauge Transformation and Gauge SymmetryGauge 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]
$\dpi{150} \bg_red \frac {dx}{dt}=u+t\frac {d^2x}{dt^2}$

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

$\inline \dpi{150} \bg_red \frac {\partial^2 \psi}{\partial x^2}=\frac {1}{v^2} \frac {\partial^2 \psi}{\partial t^2}$

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 ]
$\dpi{150} \bg_red i\hbar\frac {\partial \psi}{\partial t}=H\psi=(\frac{p^2}{2m}+U)\psi, p=\frac{d}{dx}$

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.