**Why **is Energy Conserved? — July 01, 2012.

Manmohan Dash,

**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*, **笑**, **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. **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 **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/dt**) **will** 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 **p**–**dot**, 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 **E **= 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, **V **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 **gravity**; *gravity 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

towrite in the line ofdefinition: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)

U= – int [F.dx],energy (in two useful forms)

L=T–U,H=T+UAction (gets defined from “lower” energy; in two forms, as an integral) The End of Physics Formalism.

Lower: differential, Higher: Integral.

where **U **= **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; **U **= 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, **a **= **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) **E **= 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** = **m**–**c**–**squared**, 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. **time**. *Energy 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 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: https://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.

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