Conservation of Energy
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So far we have obtained two of the big three conservation laws of physics. Deduced from the relation between Reality and Nothingness, they are absolute; that is, they admit of no exceptions. Now we come to energy, whose conservation law is more contrived than deduced and whose conservation law admits of at least one grand exception.
In our study of the calculus we learn a simple calculation that physicists have used to define energy as a conserved quantity. If the argument of an integral equals zero, then the indefinite integration yields an undefined constant. Conservation laws are formal statements that certain things remain constant, so if we can find some suitable zeros in our description of Reality, then we may contrive conservation laws by integrating them.
Newton's second law of motion gives us the textbook example of a suitable zero. In the standard derivation we equate the force applied to a body to the rate at which the body's linear momentum changes and then we re-express that equation by subtracting the applied force from the rate of momentum change to obtain the desired zero. We integrate that re-expressed equation over (with respect to) the distance that the body crosses while the force acts upon it and we use the vector dot (or inner) product to ensure that the integration includes only the component of the applied force acting in the direction parallel to the motion of the body. What we get looks like this;
∫(mdv/dt - F)dr = ∫0dr = E,
in which equation E (for Energy) represents the constant of integration.
Before we go any further I want to integrate the first term of that equation outright. I can do that because the pseudo-infinitesimal distance that the body crosses equals the product of the body's velocity and the pseudo-infinitesimal interval of time elapsed in the crossing; that is, we have dr = vdt. When I make the appropriate substitution into Equation 1 I obtain
mv2/2 - Fdr = E.
We recognize the first term in that equation as representing the kinetic energy of the forced body and the second term as representing the forced body's potential energy relative to the body (or collection of bodies) exerting the force upon it. Our derivation of the equation tells us that the sum of the body's kinetic and potential energies does not change as the body moves.
Does that equation represent a true conservation law? We should entertain some skepticism here. After all, we know that we don't actually describe any given body's energy with a number that never changes. Does Equation 2, then, give us merely the illusion of a conservation law? We have only one way to find out the answer to that question: we must try to devise a way to break the law.
I used two-body interactions as the first step in testing Equation 2 as a conservation law. We already know that the bodies will exert upon each other forces that are equal and oppositely directed along the straight line passing between the bodies. We have a version of Equation 2 for each of the bodies, so we may add those equations together to obtain a description of the total energy possessed by the two bodies together. We have
Etotal = E1 + E2 = m1v12/2 + m2v22/2 - F21dr1 - F12dr2.
Consider the situation in which the common center of mass of the two bodies does not move. This is the situation in which the net linear momentum of the two bodies equals zero and, thus, in which we have
m1v1 = -m2v2 and, therefore, m1dr1 = -m2dr2.
Using those proportions, we may rewrite Equation 3 as
which means that at all times the total energy of the system equals 1+m1/m2 times the sum of Body-1's kinetic and potential energies. We recognize that equation as just a slightly fancier version of Equation 2 and infer that the total energy of our two-body system does not change.
But an equation considered all by itself tells us very little different from nothing. Consider, then, a more explicit description of the two-body encounter: both bodies move directly toward their common center of mass and the force that each exerts repels the other in this particular example. At any given instant each body's kinetic energy and its potential energy relative to the other body add up to a total energy that does not change throughout the encounter; that is, T1+U1 = E1 and T2+U2 = E2. Proof and consequent verification of that proposition comes from a closer examination of the encounter between the two bodies.
In any interval of time elapsing while the bodies move toward each other each body's potential energy increases by some amount and the body's kinetic energy decreases by the same amount. At some instant, the same instant for both bodies, the bodies' kinetic energies come equal to zero, the bodies become motionless, and then the bodies begin to move away from each other. At any amount of time after that instant each body possesses the same kinetic energy and the same potential energy that it possessed the same amount of time prior to that instant, so the distribution of the bodies' energies forms a perfect symmetry about that instant.
We could conceivably spoil that symmetry (and thus possibly enable a violation of the conservation law) in one of two ways: either one body can give some of its energy to the other body or the force between the bodies may undergo an inherent change, thereby changing the temporal profile of the energies possessed by the bodies. If one body gives some of its energy to the other, then at some given time after the instant of motionlessness one body moves slower than it did and the other body moves faster than it did at the same time prior to the instant of motionlessness. But that outcome violates the law mandating conservation of linear momentum, so we may say with certainty that energy transfer cannot happen in this situation. As for an inherent change in the force acting between the bodies, that also cannot happen, but the proof and verification of that proposition must wait for the next section.
Let's extend our analysis of a two-body encounter further by asking how an observer moving with a speed V relative to the system's center of mass would calculate the energy of the system. As a start the moving observer would measure the bodies' velocities as v1' = v1+V and v2' = v2+V, which makes the total linear momentum of the system in the V-frame
P = m1(v1+V) + m2(v2+V) = (m1+m2)V;
that is, the system's net linear momentum equals that of a single body of mass m1+m2 moving with velocity V. In order to calculate the total energy of the system the moving observer adds together the results of reintegrating the force equation for both bodies. Reintegration of the force equation for Body-1, m1dv1'/dt - F21 = 0, yields
in which equation F21dt = m1v1. Adding that equation to the reintegrated force equation for Body-2 yields
which differs from Equation 4 only by the additional term ½(m1+m2)V2.
In the previous example I claimed that the bodies' encounter cannot happen in a way that has one body exiting the encounter with more kinetic energy than it had when it entered the encounter and the other body exiting with less. So long as Reality upholds that claim then the system conserves energy. But that claim does not hold up in this example. Did I go wrong, then, in making my original claim? And, if so, can we manipulate this example in a way that increases or diminishes the total energy of the two-body system, thereby invalidating my presumed conservation law? If we can answer yes, then we have serious logical trouble on hand, because this example merely shows us how a moving observer sees the previous example.
Suppose that the center of mass of this two-body system appears to us (if we could somehow mark it) to move in the same direction in which Body-2 initially moves. Assume also that we begin to observe this system when the two bodies are so far apart that their potential energies are negligible; that is, the potential energies are so small relative to the bodies' total energies that we can ignore them as if they were equal to zero. Thus the two bodies start out with energies that are almost entirely kinetic; that is,
After some time elapses we observe Body-2 come to rest and begin to move in the opposite direction, even though we see its distance from Body-1 continuing to diminish. Shortly thereafter the distance between the bodies stops decreasing and begins to increase. Then we see Body-1 come to rest and begin to move in the opposite direction. Later yet, when the bodies have gone as far apart as they were at the beginning of our observation of them they have the kinetic energies
You may be tempted, as I once was, to see in those equations a potential for the creation or destruction of energy, but that temptation evaporates when you recall that in this experiment, as I have described it, m2v2 = m1v1. If you use that equation to make the appropriate substitution in either Equation 10 or Equation 11, you will see that Body-2 gains exactly as much energy as Body-1 loses in the encounter. We can easily see how the transfer of energy took place. At first Body-2 moves against the force acting to drive Body-1 and it apart, thereby converting its kinetic energy into potential energy. After the instant when it comes to rest, Body-2 moves with the force, converting potential energy into kinetic energy, while Body-1 continues to move against the force, still converting kinetic energy into potential energy. After Body-1 comes to rest and begins to move with the force the amount of potential energy available to it for conversion to kinetic energy has been so depleted by Body-2 that Body-1 has less energy available to it than it would have had if both bodies had come to rest simultaneously.
Equations 10 and 11, vis-a-vis Equations 8 and 9, reflect a clear image of the process that NASA uses to send space probes deep into the outer solar system by bouncing them off Jupiter's gravitational field. That fact illuminates the contrast between the two situations that I have described above. For an observer occupying a non-rotating jovocentric frame of reference Voyager 1, for example, left the Jupiter system with as much kinetic energy as it had when it entered the system, albeit expressed in a different direction. On the other hand, for an observer occupying a non-rotating heliocentric frame of reference the probe left the Jupiter system with more kinetic energy than it had when it entered the system, enough more to carry it out to Saturn and beyond.
Thus it seems as if our energy equation does, indeed, express a conservation law. But before we can claim that statement as fully true to Reality we must address a possibility that I have not yet considered - what happens when the force changes with the elapse of time? As we contemplate that question we will find that, unlike the other two fundamental conservation laws, we cannot call this one absolute. I can only assert it as fundamental by way of the mysteries of time travel as described by Herbert George Wells in his famous 1895 novel.
On Timeward Force
In the foregoing discussion I have assumed that the force exerted between two bodies cannot be expressed legitimately as an explicit function of the elapse of time. Of course, the force acting on a body can vary as time goes by, but that variation comes entirely from motion of the body through a region of space in which the force varies with position, even when, as we saw above, the motion of the force generator makes the force at any given point change with the elapse of time. I have taken as my proposition the statement that the rate at which the force acting on a body moving with velocity v = (vx, vy, vz) changes conforms to the mathematical statement that
dF/dt = ∂F/∂t + (∂F/∂x)vx + (∂F/∂y)vy + (∂F/∂z)vz,
in which equation we have as true to Reality at all times that ∂F/∂t = 0.
In order to prove and verify that proposition I must make a digression. I want to take a quick look at the theory of time travel and see how it relates to conservation of energy.
Imagine looking down on a symmetrical smooth-sided hill whose apex lies above the origin of an x-y plane. Our man Sisyphus rolls a large spherical rock up the hill along the x-axis. Interaction between the rock's weight and resistance from the hillside generates a force that pushes the rock parallel to the hill's slope. Working against that force as he rolls the rock up the hill, Sisyphus produces a potential energy that we must describe mathematically as an explicit function of the x-coordinate of the rock. Just as Sisyphus gets the rock to the top of the hill a wedge on the ground gives the rock a slight shove in the y-direction and the rock rolls along the y-axis, gaining speed as the force acting on it converts potential energy into kinetic energy. With a sigh Sisyphus begins the long walk downhill to retrieve his rock.
Our knowledge that work done in the x-direction could yield a force oriented in the y-direction comes from the shape of the hill and the theorem of generalized force. If we have a coordinate q and if we have the potential energy of a body given to us as an explicit function of q (U = U(q)), then the force acting on the body,
Fq = -∂U/∂q,
will push the body in the q-direction, whatever it may be. In our example we must have the rock's potential energy as a function of the rock's altitude, of its height above some arbitrarily determined level in the landscape surrounding the hill. But we describe the rock's altitude on the hill with a function of its x- and y-coordinates, so, while the fundamental force coming from the rock's potential energy acts in the vertical direction, the shape of the hill gives it expressions in the x- and y-directions. So Sisyphus can give the rock some potential energy by pushing it in the x-direction and the expression of the force in the y-direction can then confound him just when he believes that he has finally brought his burden to the top of the hill. But if, instead of a hill for this example, I had used a ridge of uniform height whose crest ran along the y-axis, then we would have had no changes of altitude in the y-direction; the potential energy function for the rock would have had no explicit dependence upon the y-coordinate; and no force would have acted in the y-direction.
Now suppose that we have a body with a potential energy whose mathematical expression contains, in addition to other coordinates, an explicit (that is, a direct) reference to elapsed time. In that case we would have a force
Ft = -∂U/∂t ≠ 0
pushing the body in the temporal direction. But what does it mean to say that we have a force exerted in the direction of time?
When we exert a force in a certain direction, the body upon which we exert the force accelerates in that chosen direction. But we know that acceleration changes the body's velocity, the ratio of distance crossed to time elapsed. Timeward force should impose a temporal acceleration upon a body, but what could we possibly mean by the ratio of time crossed to time elapsed?
We often describe time as a river, one whose flow carries us past events as the flow of water carries us past places. We even have a way of measuring distances between places on the banks of the River Chronos: we use events, other places on the banks, that we have contrived to position with uniform separations and count those as distance; that is, we use clocks. Now, to mix metaphors very badly, suppose that I sit on my raft with a clock that counts one hour for every sixty minute markers that I pass on the riverbank. With reference to that image it makes sense for me to say that I move through time at a speed of sixty minutes per hour. I rescue that statement from utter silliness by pointing out that in the same image a person going down the river in a motorboat would pass more than sixty minute markers for every hour elapsed on their clock. We thus discover that we can indeed define a timeward velocity if we take the ratio of the times measured on two different clocks: we would measure time crossed on a clock set up on a stationary base and measure time elapsed on a clock mounted on a time machine.
And a time machine, of the kind described by H. G. Wells, is exactly what the force described in Equation 14 would enable us to build. Once we have built such a vehicle, we need only flip a few switches and push the velocity bar forward in order to embark on a grand adventure into ages yet to come. Like Wells's Time Traveler, we would watch future history flash by in a hyperfrenetic blur.
On the other hand, if we were observing from outside the time machine, we would see everything aboard the time machine slow down drastically. If our Time Traveler were to move at a speed of 100,000 years per hour (about 8.766 x 108 times our normal rate of travel), a little slower than Wells's Time Traveler did, then the simple act of taking their hand off the velocity bar and leaning back in their seat would appear to us to take as long as a century. And that observation exposes the error in my presumption.
In both of the movies made from Wells's novel a major component of the time machine is a large spinning disc. For simplicity's sake let's assume that the time machine must spin up the disc, giving it its full, rather large, angular momentum, before it can engage the time drive. We wave good-bye to the Time Traveler and watch as they and the time machine appear to freeze into utter motionlessness. We watch as the spinning disc seems to stop rotating altogether, its angular momentum dropping close to zero though nothing exerts a torque upon it. Likewise the linear momenta of any parts in reciprocating or other straight-line motion diminish, though nothing exerts the forces necessary to mediate the obligatory equal and oppositely directed reactions. To our astonishment our time machine seems to violate two absolute conservation laws.
Perhaps in our observation we have missed something? Perhaps the time machine obeys the conservation laws by increasing its mass to compensate the slowed motions? Such an increase would certainly save the momenta, but in order to effect it the time machine would have to violate the law pertaining to conservation of mass. We now have a contradiction: our time machine must necessarily violate at least one of the conservation laws that we have already deduced. Logic does not allow us to incorporate contradictions into the construction of formal systems and, thus, obliges us to discard at least one of the premises that led to the contradiction. At this point I cannot falsify the conservation laws that I have in hand, so I have only one premise to discard as false to Reality - my assertion that timeward force exists. I must assert instead that
Ft = -∂U/∂t = 0
forever and always. That proposition necessitates in turn that whatever potential energy that a body or system of bodies possesses cannot be such that we can describe it by an explicit function of the elapse of time.
If I add that proposition to my previous statements pertaining to kinetic energy, then I seem to have come up with a firm law of conservation of energy; that is, I think that I can state with full confidence that no phenomenon can create energy ex nihilo or destroy energy ad nihilo. But I have a nagging suspicion that I have still neglected something in this derivation, that we have not yet blocked all ways to circumvent the conservation of energy. Our definition of energy in terms of work done seems to offer some possibilities for circumventions if we can manipulate the forces in suitable ways, two in particular.
Suppose that two bodies exert each upon the other forces whose descriptions include the velocities of the bodies. We have two basic possibilities: either the force may act in a direction perpendicular to the direction of the forced body's velocity or it may act in the direction parallel to the forced body's velocity. Because I can dismiss the first possibility quickly and easily I will consider it first.
Although it's a bit of a cheat to mention it at this stage of drawing the Map of Physics, we do know a force that acts in a direction perpendicular to the direction of the forced body's velocity - the Lorentz force, the force that a magnetic field exerts upon a moving, electrically charged body. Because the force acts in a direction perpendicular to the forced body's velocity, it does no work upon the body, adding nothing to and subtracting nothing from the body's energy. This kind of force, then, has no effect upon conservation of the body's total energy and thus has no effect upon the validity of our conservation law.
If a force acts in the direction parallel to the velocity of the forced body, then it certainly does work upon the body, converting kinetic energy into potential energy and vice versa. Such a force certainly has relevance to any conservation law pertaining to energy, but what makes that force nonconservative also ensures that it cannot exist.
Like any other force, a velocity-dependent force must automatically obey the conservation laws pertaining to linear and angular momenta. Given the fact that the force conserves linear momentum, for example, necessitates that its description incorporate the velocities of both bodies involved in the interaction; that is, we must have F = F(v1, v2). Such a functional dependence provides the equal part of the action-reaction pair and the geometric factor incorporated into it provides the oppositely directed part of the action-reaction pair, exploiting the fact that we take the distance between the two bodies as positive when we measure it from Body-1 to Body-2 and negative when we measure it from Body-2 to Body-1.
If two bodies exert such a force each upon the other, then we can calculate for one of them, let's say Body-2, a description of its potential energy due to its interaction with the other body. We have U2 = U2(v1, v2). We don't know the actual mathematical form of the function, but that form does not concern us here: we need only the fact that the function depends on both v1 and v2. You see, the importance of that dependency lies in the fact that someone might manipulate Body-1 and change its velocity and in so doing change the magnitude of Body-2's potential energy. But that change is not like the change of potential energy that a body acquires when it moves toward or away from the crest of a potential energy hill or, as seen by some observers, when the crest of a potential energy hill moves toward or away from the body. No, this change gives us a change in the height and shape of the hill while the body lies on it and it makes ∂U2/∂t ≠ 0. But that makes Body-2 subject to a timeward force and the no time travel theorem that we deduced above forbids such a thing from existing, so any force, however oriented, that produces a potential energy that conforms to such a description cannot exist.
Thus we may infer that velocity-dependent forces do not invalidate our proposed conservation of energy law.
One Other Possibility
In testing the validity of the proposition that Reality always conserves energy I have considered and dismissed the existence of non-conservative forces. But we have one other factor in the definition of energy that we must consider, though it seems unlikely to us to have an effect upon the conservation law. Can we contrive ways to create or to destroy energy by means of non-conservative geometries?
Clearly we can make the energy possessed by a body increase or decrease non-conservatively if the force exerted upon the body changes inherently as the body goes around a closed path. Consider the simplest example: the body goes a distance x from point A to point B against a force F1 and then a force F2 pushes the body from point B to point A. Every repetition of that cycle creates a net energy
E = F2x - F1x.
Of course, we cannot create any such cycle, because a force that would change in the right way to make it possible would also violate the no time travel theorem.
But what would we get if the distance between points A and B were to change as the body makes its circuit? Suppose that the body goes a distance x against the force F in going straight from point A to point B and then, because the intervening space has expanded while it dwelt at point B, the body goes a distance x+a with the same force F in going straight from point B to point A. In this example we calculate the net work done upon the body as
E = F(x+a) - Fx = Fa.
We thus would gain purely created energy, coming ex nihilo and not from any other body or collection of bodies, the body generating the force included.
In devising and presenting that example I have made a tacit assumption that deserves some scrutiny. I have assumed that when the space in which I have embedded my imaginary experiment expands point B carries the body along with it. Whether I can assert that such an assumption is true to Reality depends upon the nature of space and the nature of the expansion. Our own experience of the space in which we find ourselves embedded predisposes us to believe that space has such a nature as to make my assumption false to Reality. That experience and that belief have led us tacitly to postulate that space is simply an immutable and inert vessel in which matter carries out its existence and motions. Until I find reasons that compel me to believe otherwise, I will accept that postulate as true to Reality for all of space.
However, we already know that some regions of space, however small and however temporarily, must have manifested a non-conservative geometry in order to enable the creation of matter when the Universe blossomed into existence. But we should not have that piece of knowledge in our possession at this stage of the drawing of the Map of Physics, so until reason produces the relevant propositions I shall strive to feign ignorance of it.
Encoding Conservation of Energy
At its most fundamental physics is the science of motion. For more than a century and a half the discoveries of self-sustaining forcefields and a zoo of elementary particles have obscured that fact. Those discoveries, though, were unintended side effects of studies aimed at clarifying what makes bodies move the way they do. Ørsted's discovery of the magnetic effect of an electric current, which led to Maxwell's electromagnetic theory of light, came from Ørsted's effort to explain the queer motions of compass needles in thunderstorms. Twentieth-Century physicists' discovery of the neutrino came from their efforts to understand the motions of nuclei and their emitted electrons or positrons coming out of certain beta decays. Behind the shimmering veil of those discoveries and all others in physics stands the Platonic Idea of bodies in motion.
Motion is the fundamental idea of physics. For that reason anything that sums up the legitimate motions of a system of bodies also sums up the laws of physics that apply to that system. Nicely enough, the concept of energy gives us just such a summation by way of Hamilton's Principle.
William Rowan Hamilton (1805 - 1865), an Irish mathematician, constructed his eponymous principle on the concept of virtual work that was conceived in 1717 by the Swiss mathematician Jean Bernoulli (1667 - 1748) and expanded in 1743 by the French mathematician Jean Le Rond D'Alembert (1717 - 1783). What Hamilton did looks very much like the definition of energy that I showed you earlier and in its raw form Hamilton's Principle looks like a statement of the law of conservation of energy.
Bernoulli stated that if the forces acting on a body are in equilibrium; that is, if the net force acting on the body equals zero, then the virtual work done upon the body by any minuscule virtual displacement of the body from its true position must equal zero. D'Alembert extended that concept by pointing out that the body's kinetic reaction, dp/dt, is also a force and, thus, that the forces acting on a body are always in equilibrium under Newton's laws. We can write D'Alembert's proposition in its simplest form as
∫(F - dp/dt)δx = 0,
in which δx = (δx, δy, δz) represents the minuscule virtual deviation from the body's true position x at any given instant.
Following Hamilton's lead, let's pull the variation in position inside the parentheses and get
∫Fδx - ∫(dp/dt)δx = 0.
In that equation the first term represents the negative of the virtual variation in the body's potential energy due to the virtual deviation in the body's position. It may seem that the second term represents just the corresponding variation in the body's kinetic energy, but the situation is not so simple: although the differential and the variational both represent minuscule differences and commute with each other, they do not represent the same operations, so we cannot interpret dx/dt as the body's velocity as I did in the derivation of Equation 2. We must take a more subtle approach in interpreting the second term of Equation 19.
Start by rewriting that second term as
In devising that revision of the second term I have assumed that the body's mass does not vary and I have used the upper-case tee to represent the body's kinetic energy in accordance with convention.
Equation 19 thus becomes
δT-δU-md(v•δx)/dt = 0.
That equation's third term suggests the next step that we will take, but first we need to look a little more closely at the motion that we are describing in the creation of these equations. We have tacitly taken two points, x1 and x2, and the two instants, t1 and t2, when the body passes through those two points and in imagination drawn the path that the body follows in going from x1 to x2. Around that true path we can find an infinite number of alternative paths that the body could conceivably have followed. Each of those paths differs from the true path by the virtual deviation δx(t) subject to the condition δx(t1) = δx(t2) =0, which condition simply encodes the fact that at the points x1 and x2 the alternate paths coincide with the true path. Now we can multiply Equation 21 by the increment of time dt and integrate the result to obtain
which is the mathematical statement of Hamilton's Principle. We get a zero output from this integral because we have calculated the definite integral (one with definite limits) from an integral whose indefinite form yields a constant.
For any given body we won't necessarily describe the kinetic and potential energies with functions of the Cartesian coordinates. In some cases we may find it more natural and convenient to describe them with functions of other coordinates (such as, for example, polar coordinates), so I want to look at Equation 22 in terms of generalized coordinates. We have the transformations x = x(qi) for the generalized coordinates qi, the subscript i referring to the different coordinate axes used in the system under consideration (in Cartesian coordinates we have, for example, q1 = x, q2 = y, and q3 = z), so we thus have
T = T(qi, wi)
U = U(qi),
in which wi = dqi/dt represents the generalized velocity. To show that we have ensured conformity of the functions representing energy with the no time travel theorem, we must write those functions in such a way that neither makes explicit use of time as a variable and write the potential energy function in addition in a way that excludes explicit dependence upon the generalized velocities.
Because variation of a function works the same way as does differentiation, we may write out the variations of the energy functions as
δT = (∂T/∂qi)δqi + (∂T/∂wi)δwi
δU = (∂U/∂qi)δqi.
In those equations I have tacitly used the Einstein convention, which tells us to use all values of the index i when it appears in both the numerator and the denominator in a given term.
Variation and differentiation commute with each other (which means, we may apply them in either order), so we can write δwi = dδqi/dt. That substitution, along with Equations 25 and 26, transforms Equation 22 into
We may integrate the second term of that equation by parts and obtain
Using that result to make the appropriate substitution, we can rewrite Equation 27 as
which equation we can assert as true to Reality if and only if
We have thus obtained the Euler-Lagrange Equation, which equation transforms a description of a body's kinetic and potential energies into the equations of motion whose solutions comprise a complete description of the body's motion. We have as true to Reality, then, that a description of a body's energy distribution encodes the laws of physics that apply to the body in a given situation.
As an example consider a body whose potential energy we describe with a function that reflects a purely spherical symmetry. In this case we want to use polar coordinates to describe the body's location and motion, using r to represent the body's radial distance from the center of the symmetry of the body's potential energy and to represent the body's angular position in the counter-clockwise direction relative to an arbitrarily chosen radial line. For convenience I shall assume that the body moves only in the coordinate system's equatorial plane. In that system we have for the generalized components of the body's velocity v = dr/dt and ω = dθ/dt, so that we have for the energies
U = -K/r,
in which equation K represents some constant factor, and
T = 1/2mv2 + 1/2mr2ω2.
In this example we must apply the Euler-Lagrange Equation twice, once with qi=θ and wi=ω and then again with qi=r and wi=v. We obtain from the first application
which automatically states the law pertaining to conservation of angular momentum. And we obtain from the second application
That equation, combined with Equation 33, describes a body moving on a path that traces a conic section (a circle, an ellipse, a parabola, or an hyperbola) whose focus lies at the center of the coordinate frame. Note how the second term in Equation 34 automatically brings a description of the centrifugal force into play.
At last, then, we have obtained two major results. We know that energy can be neither created nor destroyed, but can only be transformed and transferred; that it is subject to a conservation law, albeit one that admits exceptions under circumstances that do not occur in normal space. And we also know that a description of the distribution of energy in a system of bodies encodes the laws of physics that govern that system of bodies.
Oh, By The Way
I noted above that I obtained the inspiration for an important part of my derivation of the conservation of energy theorem from a contemplation of Herbert George Wells's 1895 novel about a man who travels to the year A.D. 802,701 and beyond. I showed you that the machine that Wells described could not possibly work because its operation would violate the absolute conservation laws pertaining to linear and angular momenta, but some years ago I conceived three other reasons why Wells's Time Traveler could not have made his journey into Futurity. Shortly after beginning his journey the Time Traveler guesses that the machine is carrying him through time at the rate of about one year per minute, about half a million times faster than normal. I'll use that speed in what follows here:
I. From the Time Traveler's point of view everything in the world beyond the time warp surrounding his machine seems to speed up. That would include the vibration rate of photons. Because we can relate a photon's vibration rate to its energy content by way of Planck's formula, we can calculate that the Time Traveler would see visible light, whose photons have energies ranging from 1.6 electron-volts to 3.2 electron-volts, transformed into gamma rays with energies in the range from 800,000 electron-volts to 1,600,000 electron-volts. Further, he would see the energy in sunlight, which normally strikes Earth at a rate of about one kilowatt per square meter, intensified to 500 megawatts per square meter. Such a bombardment would strip every atom in the Time Traveler, his time machine, and the air in the machine down to its nucleus and pervade the resulting plasma with a thick electron-positron haze.
Looking at that phenomenon, we can see that the time machine seems to be enveloped in a remarkably stiff gravitational field. That means that not only does it subject light coming into the machine from outside to a strong blue shift, but it also subjects light originating in the incandescent bulbs inside the machine to a strong red shift. Because the time machine allows the operator to control that effect, it allows the operator to create energy ex nihilo or destroy it in a way that we cannot do with true gravity, which does not change. Energy carried in the time machine's batteries and converted into light in the bulbs would disappear from Reality as that light crossed the boundary of the time warp. This gives us another example of how a time machine would violate the law of conservation of energy.
II. Everything in the world seems to speed up in the Time Traveler's view. That would include the motions of the molecules comprising the laboratory and the air within it. Tragically for the Time Traveler, the temperature of a body is proportional to the average kinetic energy of its molecules, which energy is, in turn, proportional to the square of the relevant speed of molecular motion. If the speeds of the molecules increase by a factor of half a million, then the temperature of the body they comprise increases by a factor of one-quarter trillion. We can only guess at what the temperature was in the Time Traveler's laboratory, but a good guess would be 293 Kelvin (20 Celsius or 68 Fahrenheit). Boosted by the time machine's operation, that temperature would appear to the Time Traveler to have gone to 73.25 trillion Kelvin, a temperature typically found near the center of a supernova in the first instants of its explosion. He would have had little time to contemplate that phenomenon, because the pseudo-gravitational effect of the time warp would almost instantly bring a large quantity of the superhot air into the time machine.
III. Sitting in the saddle of the time machine the Time Traveler saw Mrs. Watchett, walking sedately, cross his laboratory "like a rocket". Had she knocked something off a table, he would have seen that something leap to the floor as if shot from a gun. Gravity would appear greatly stiffened, by a factor of one-quarter trillion because we measure it in accordance with the inverse square of the elapse of time. Unless the time machine included a suitable amount of adjustable Cavorite, that gravity would have pervaded the machine with tragic result.
Blasted, burned, and smashed flat, Wells's Time Traveler would not have gone far. But that's only a criticism of "The Time Machine" taken as science fiction. I still enjoy the story as science fantasy. When I'm not exploring the physics implicit in the story I quietly contemplate the grand vista of Time that Wells laid out and I find myself haunted by the spectral images of the effete Eloi and the sinister Morlocks, ghosts of futures past and, I hope, not of futures yet to come.
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