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My physics instructor told the class, when lecturing about energy, that it can't be created or destroyed. Why is that? Is there a theory or scientific evidence that proves his statement true or false? I apologize for the elementary question, but I sometimes tend to over-think things, and this is one of those times. :)

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    $\begingroup$ "...but I sometimes tend to over-think things, and this is one of those times." You should study physics then. $\endgroup$ – Nikolaj-K Jan 7 '12 at 13:15
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    $\begingroup$ Possible duplicate of Is energy really conserved? and links therein. $\endgroup$ – Qmechanic Feb 1 '12 at 20:37
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  • At the physics 101 level, you pretty much just have to accept this as an experimental fact.

  • At the upper division or early grad school level, you'll be introduced to Noether's Theorem, and we can talk about the invariance of physical law under displacements in time. Really this just replaces one experimental fact (energy is conserved) with another (the character of physical law is independent of time), but at least it seems like a deeper understanding.

  • When you study general relativity and/or cosmology in depth, you may encounter claims that under the right circumstances it is hard to define a unique time to use for "invariance under translation in time", leaving energy conservation in question. Even on Physics.SE you'll find rather a lot of disagreement on the matter. It is far enough beyond my understanding that I won't venture an opinion.

    This may (or may not) overturn what you've been told, but not in a way that you care about.

An education in physics is often like that. People tell you about firm, unbreakable rules and then later they say "well, that was just an approximation valid when such and such conditions are met and the real rule is this other thing". Then eventually you more or less catch up with some part of the leading edge of science and you get to participate in learning the new rules.

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    $\begingroup$ What advantage do we get by recasting the question in terms of time translational invariance? $\endgroup$ – Revo Jan 7 '12 at 14:28
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    $\begingroup$ @Revo: You're able to understand this rule as one of a class of conservation principles that arise from symmetries so you feel a whole lot smarter. $\endgroup$ – dmckee Jan 7 '12 at 16:33
  • $\begingroup$ Also, I would argue that translational invariance is much easier to understand intuitively. It is no problem to imagine doing an experiment now and at a different point in time, whereas energy is a more abstract concept. $\endgroup$ – scaphys Mar 2 at 12:23
  • $\begingroup$ For a $101$ level understanding, the Feynman lectures have a darn good explanation: feynmanlectures.caltech.edu/I_04.html. This approach also paves a natural way towards generalizing what we call energy, for example, it becomes easy to understand why we should give the electromagnetic field energy just as much validity as the kinetic energy of a particle. $\endgroup$ – Dvij Mankad Mar 2 at 14:05
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In a small area with Earth's gravitational field, general relativity very closely resembles special relativity. For low speeds, special relativity closely resembles Newtonian physics. If we define the total kinetic energy of an object to be the triple integral of the product of its density and half the square of its speed and its total momentum to be the triple integral of the product of its density and velocity, it can be mathematical proven that in any system where momentum is conserved, the change in total kinetic energy is the same in all frames of reference. It turns out that when ever a system with no external force loses kinetic energy, it gains the same amount of thermal energy. Gravitational potential energy of an object can also be defined as mgh. Using the equation (vf)^2 = (vi)^2 + 2ad, we can derive that the the sum of the total potential potential energy and kinetic energy of an object in free fall stays constant. Centrifugal force is a fictitous force so there is no centrifugal potential energy how is energy conserved when you do work to move an object closer to the center of a centrifuge in outer space? It's because it has another fictitous force called the Coriolis force and when you're moving it closer to the centrifuge, it exerts a reactive Coriolis force on it increasing its rate of spinning and therefore giving it more kinetic energy.

I believe it has been proven that there is exactly one way of defining relativistic mass and momentum as a function of rest mass and velocity and given two objects of any rest mass and velocity, what rest mass and velocity the system will have if they collide and combine such that

  • Total relativistic mass and momentum are conserved
  • The relativistic mass is equal to the rest mass at zero velocity
  • The momentum is zero at zero velocity
  • The rate of change of momentum with respect to velocity at zero velocity is equal to the rest mass
  • In any frame of reference, two objects of a given rest mass and velocity if they combine will combine into a system with the same rest mass and velocity

and that each object has a rest mass and actually follows those laws in our universe.

In the real situation of a system with no external forces in the absense of a gravitational field, we can define the rest mass of a spinning object to be its relativistic mass in the frame of reference of its velocity and when two objects combine, they will either gain thermal or spinning kinetic energy and it's actually the thermal energy and spinning kinetic energy that contributes to its rest mass but two objects of a given rest mass and velocity will combine into a system of the same rest mass and velocity regardless of the source of their original rest mass.

I believe that in general relativity, energy and momentum are not necessarily conserved. I think an electromagnetic field doesn't curve space time and does not influence a gravitational field in any way other than accelerating a charged particle which in turn affects the gravitational field with its gravity and in the absense of particles, an electromagnatic field will not affect a gravitational field one speck. That means an electric field will not accelerate a charged black hole. Maybe a charged black hole could undergo a hyperbolic orbit with a charged particle and the charged black hole will accelerate the charged particle with its mass and charge but the charged particle will accelerate the charged black hole only with its mass and not with its charge, creating a net change in momentum and if it doesn't satisfy an exact condition, also a net change in total energy.

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I would argue that conservation of energy is part of its definition, in other words energy was designed so as to be conserved. Because quantities that are conserved can reasonably be regarded as convenient. Think of it this way. Let's say we're a few hundred years back and we're working on this emerging concept we decided to call 'energy'. We probably already had the sensible idea of associating energy with motion (calling it kinetic energy) and temperature (thermal energy), maybe even in such a way that when a moving object is slowed down by friction, the heat given off and the loss in velocity correspond to equal amounts of this 'energy'. Later in time we would come to realise that we hadn't thought it all the way through. Say I have a sliding object on a curved rail that starts off vertically at the top and ends horizontally at sea level. I decide to hold the object still at a certain height. It is initially motionless at room temperature. I let go, it slides down the rail and heats up until friction brings it to a stop somewhere on the horizontal component of the rail. It is now motionless, as before, but hotter. It therefore has the same amount of kinetic energy but more thermal energy. Where did it come from ? Placing an object above the ground gives it the potential to gain motion or to produce heat, or rather, height seems to be convertible into heat and motion. Therefore it seems reasonable to assign energy to height. Need a name for that form of energy ? Call it 'gravitational potential energy'. According to me, the concept of gravitational potential energy was invented in order to account for this otherwise 'gain' of energy. Indeed, it enables to claim that the increase in temperature results from the simple conversion of already existing energy.

The next step involves quantitatively defining our newly invented form of energy. Having already quantitatively associated energy with heat and motion, we could experimentally determine the amount of energy that a 1kg object has when held 1m above the ground by letting it slide down the rail and measuring the energy converted into forms we are already familiar with. Repeating with different masses at different heights we could derive a formula for gravitational potential energy that satisfies the sought after conservation of energy.

I have no idea which order the various formulae actually came up in, but I strongly believe this must have been the reasoning behind it.

A more mathematical approach based on the same idea is as follows. Assume you've defined (macroscopic) kinetic energy to be $\frac{1}{2}mv^2$. Using Newton's 2nd law of motion, you could prove that when an object is moving through a constant gravitational field $g$ and is subjected to no other force, the quantity $\frac{1}{2}mv^2+mgh$ is conserved. In your search for a quantity that is conserved, you would then define this whole thing as the energy of the object in question and, having called the first term kinetic energy, you would name the second one "gravitational potential energy".

In other words, energy is a mathematical concept, defined and scaled so as to be conserved in isolated systems. By the way, note that an isolated system is merely defined as a system where energy is conserved, so leaving out gravitational potential energy would result in a marble rolling down a slide inside a closed and thermally insulating box no longer being called isolated, and one might then consistently say that gravity is adding energy into the system. Whether this view is any less valid I don't know, but it's not the accepted one.

Anyway, why bother with such an invention ? Because to some extent it enables to predict the future. You won't actually have to drop a 75kg object along a sliding rail from 390m above the ground to know how much kinetic and thermal energy will be produced (at least in total).

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In general of course energy is not conserved, since any motion is associated with acceleration and that produces radiation that travels away from the interaction- thermal radiation from all bodies is an example. But in an isolated system energy is conserved and there is a simple classical explanation for this. Take one isolated system, you find that an important property of the space is that it is not possible to move one mass without moving an equal mass the same distance and in an opposite direction. That is sum(m.ds)=0 along any line in the direction s. We know that this is correct because a differentiation w.r.t time for constant m gives sum(m.v)=0 along any line and this is conservation of momentum- known to be true all the time. A second diff. w.r.t time gives sum(m.a)=sum(f)=0 along any line. This is the action and reaction law of forces and also known to be correct all the time. Conservation of angular momentum follows immediately and conservation of energy too.. since E=ingetral(F.ds)=int(m.a.ds)=int(mv.dv)=.5 m v^2, for constant m and noting that v=ds/dt by definition.

So, conservation of energy is a consequence of conservation of momentum. We note that energy is more useful than momentum in problem calculations as energy is just a number, while momentum is a vector and can (apparently) go to zero sometimes. Why should momentum be conserved should be taken as 'given'.. a property of space, like the sum of angles of a triangle 180 degrees and like the ratio of the circumference to diameter of a circle in general 3D space.

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protected by Qmechanic May 23 '16 at 4:49

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