Formation of equations..?

Question

I read about how certain equations which we use conveniently now came about, with equations substituting into more equations and so on. Is there any reason in physics why some equations are altered in certain ways and substituted in another? Or is the formation of equations merely a process without reason?

Answer 1

No, the formation of equations IS NOT merely a process without reason!

There's always a goal in substituting the CONSEQUENCE of one equation (NOT "the equation" itself --- you don't stick multiple "=" signs into some other equation!) into another equation. That goal may be one of several. It can be to eliminate a subsidiary variable in terms of something more fundamental, or (in physics), perhaps more directly measurable. It can be looking for deeper meaning, or connections or implications not so clear before.

I like to think of this as "marrying equations together." You do it in the hope that the children of the marriage will have some potential that was only faintly present in the parents.

Admittedly, there is some art in doing this, but reason is used, with lots of prior experience as a guide.

AN EXAMPLE: SEVEN STEPS TO THE LAW OF GRAVITY

Let me give you a somewhat simplified historical example.

After a decade or more manipulating the data he'd inherited from the tyrannical Tycho Brahe, Kepler was overjoyed to discover his "Harmonic" or "Third Law" of Planetary Motion (K3):

P^2 prop. to R^3, ..........(1)

where P is the period of a given planet orbiting the Sun, and R is the radius of its orbit. (O.K., part of the simplification is that Kepler had already found that orbits were elliptical and instead of 'R,' it was really 'a,' the semi-major axis of the ellipse, in this relationship. No matter; this is not the place for a scholarly discourse.)

So, idealizing it a bit, how did this observation help lead to Newton's "Universal Law of Gravity"?

Newton had the goal of showing that the same dynamics governed both what was happening "in the heavens" and on Earth. (Galileo's assumption was different; for example, "linear inertia" on Earth became "circular inertia" in the solar system.)

In other words, Newton was after UNIVERSAL "Laws," not just capricious, arbitrary "Laws" tailored to each specific environment.

So the question was, "Kepler's Third Law (K3) appeared to link all the planets together somehow. Did it imply something more fundamental or useful about what CAUSED that linkage, and what's more, that illustrated the same Earthly dynamical laws of motion operating out in the solar system?"

Framed that way, the path to the inverse-square Law of Gravity is REMARKABLY SHORT:

Newton had finally worked his way to the realization that if you are given, as a fact, that something is moving in a circle, with speed v, its INWARD or CENTRIPETAL ACCELERATION is:

accn. = v^2 / R. ..........(2)

(It had taken him a long and at times resisted route to free himself from the Huyghens idea of "outward endeavour," or "centrifugal force." The latter concept held up Newton's clear thinking on this matter for more than a dozen years. He didn't shake himself free from thinking in this misleading way until no less than Robert Hooke --- his despised rival --- actually asked Newton what he thought of planets being attracted and therefore being deviated INWARDS towards the Sun --- Hooke's own idea, published somewhere where Newton hadn't yet seen it. Newton actually replied, confessing that he'd never thought of it that way before --- his letter is preserved in Trinity College's Wren Library in Cambridge, England. [I've been allowed to see it and read it, very carefully --- while avoiding to touch it!] It is because of knowing what problems this concept gave Newton that I still shake my head at the notion of "centrifugal force" being taught to every high school Tom, Dick, Jane and Harry without careful pointing out of its pitfalls, when it is or is not an applicable thing to employ as an artificial calculating method rather than a real, physical fact, what its limitations are, etc.)

Enough of that. Back to the (short!) derivation.

K3 doesn't mention "v"; but it does mention P. So, simply express 'v' in terms of R and P:

v = 2 pi R / P. ..........(3)

(Or: orbital speed = circumference of orbit divided by time taken in one orbit.)

Then accn. = v^2 / R = [2 pi R / P]^2 / R = 4 pi^2 R / P^2. ........(4)

But Newton's own Second Law of Motion (N2) was that:

F = m accn. ..........(5)

where m is the mass of the thing accelerating, and F is the force that takes.

So, F = 4 pi^2 R m / P^2. ........(6)

JUST LOOK AT THAT --- IT"S GOT "P^2" SITTING IN IT, RIGHT THERE! It's just CRYING OUT to have Kepler's "P^2" (from K3, eqn. (1)) SUBSTITUTED into it. Let's do that. Then:

F = 4 pi^2 R m / P^2 = (proportional to) 4 pi^2 R m / R^3,
i.e. F is proportional to 4 pi^2 m / R^2. ........(7)

Thus, using a re-expression of the (inward) centripetal acceleration (eqn. (2)), N2 and K3 together imply a truly fundamental law of Nature,

"THE LAW OF GRAVITY" (first indication; see later remarks in the square brackets immediately below): F is proportional to the mass of the attracted object, and inversely proportional to the square of the distance.

[Newton would go on to extend this discovery to mutually attracted objects (rather than an immovable central attractor, implied in the above simplified analysis) with both masses now present in the law, moving in mutually elliptical orbits (not circular), in the end deducing an explicit form for the mysterious K used when writiing K3 as P^2 = K a^3. He also tested his formula for K by comparing results for Jupiter's satellite system with the solar system, FINDING and then CONFIRMING thereby that Jupiter's mass must be about one thousandth that of the Sun --- truly a major PEAK in mankind's understanding of the physical Universe around us. But all of that followed this simple beginning.]

I hope I've convinced you that reason --- and not just sheer dumb luck --- played a pivotal role in the above process.

I also hope that, by going through the above steps carefully, in order, you can see for yourself how the successive substitution steps are motivated throughout. Best of luck!

Live long and prosper.

Answer 2

Nothing in mathematics is done without reason or logic. This is done for several reasons.

For example Newton's famous gravitational equation:

F = GM1M2/r^2

This equation came from simplifying the more complicated field equations, so that people could use them to learn and understand more about gravity, without having to know calculus and differential equations.

So, it is done to make a more manageable equation.

Also, consider this:

Vf^2 = Vo^2 + 2ax

You can take that equation along with F = ma, and come up with:

Fx = 1/2mVf^2 - 1/2mVo^2

Which is the Work-Energy Theorem equation.

So, you took a simple kinematic equation, and turned it into a profound equation that equates work to the change in kinetic energy of an object.

In that respect, it allows us to have a better understanding of work and energy, and allows us to solve for a situation without having to know the acceleration.

Sometimes when things are rewritten in other ways, it can give us insight that we would never have dreamed of.

For example, if you take the relativistic Energy equation with the relativistic momentum equation, you can derive the following:

E^2 = (mc^2)^2 + (pc)^2

That might not look like much, but if you set the mass equal to zero, you get:

E = pc

This is a very profound result! This says that a massless particle can have momentum (given by the variable p). This comes from relativity, and predicted something that is a cornerstone of quantum physics, which wasn't discovered for 20 years or so later.

Anyway, I hope that helps explain why equations are manipulated.

Answer 3

There are many reasons why we substitute various equations at different times. Sometimes, we know one set of variables but are trying to find a solution which is expressed as a second set of variables. By carefully replacing what we do not know with what we know, we can find the answer. There are yet other times when the substitution of certain equations makes things simpler (read: more compact notation) to deal with.

In the end, derivations and equation substitution are usually not arbitrary, but serve very specific purposes which depend upon the particular circumstances of the situation.

Answer 4

You are not perfectly correct. What does H_2+Cl_2 actually mean ?... If it's H2 + Cl2, then you're near enough. 2 x H(+1) ions... = 1 diatomic molecule of hydrogen as covalent H2+ 2 x Cl(-1) ions... = 1 diatomic molecule of chlorine as covalent Cl2-. We now have (H2+) + (Cl2-) = 2HCl (Electrically Neutral) ( = 2 molecules of Hydrochloric Acid Gas (HCl(g). As Hydrochloric Acid Liquid, (HCl gas is dissolved into pure water to produce HCl(aq)).