Why do most formulas in physics have integer exponents?

[jbriggs444]

Have you thought about units? What would the units of m^0.123a^1.43 be?

kg^0.123meters^1.43/sec^2.86

Try it. If you convert a measured numeric value of mass expressed in kg to a corresponding numeric value expressed in pounds (multiply it by factor of 2.2), by what factor will the computed value of m^0.123 change?

 

[Quantum Defect]

I think the original poster is asking the question why so many of the laws of physics are "simple." I believe that s/he gets that this is what is true, based upon experiment i.e. doubling the mass for a constant force will halve the acceleration, but s/he wonders why everything works out so cleanly for us.

i.e. why is Coulomb's Law: F = constant * q_1*q_2/R^2 -- why not more gnarly exponents? Is there some deeper reason for why it all works out so prettily for us?

In the old days, an acceptable answer for most people (including Newton) would have said that the simple exponents reveal to us the hand of god.

 

[TeethWhitener]

Wikipedia has a pretty good illustration of the connection between the inverse square law found in Newton's law and Coulomb's law and the geometry of 3 dimensions:
http://en.wikipedia.org/wiki/Inverse-square_law
In fact, it doesn't have to be this way. Coulomb's law, e.g., could have the form $$F \propto \frac{1}{r^{2+\epsilon}}$$ where $\epsilon$ is some tiny number. In fact, that will be the case if the photon has a mass. The fact that we observe that $\epsilon \approx 0$ is one verification that, if the photon does have a mass, it is very very small. See the first chapter of Jackson's electrodynamics book for a better explanation than I could possibly hope to give.

 

[David Nassau]

...
I think your question lacks depth. Learn why the formulas are what they are. Continue your studies to find situations without whole number coefficients and powers. Check out the universal gravitation equation. The gravitation constant is not a "nice" number.

Your answer is condescending. It's actually a very deep question. Why are the laws of nature such that they can be described with simple math? There's not an obvious reason why this should be the case.

 

[zahbaz]

I think a good read, if you haven't done so yet, would be Feynman's lectures. He often addresses these issues.
Lectures 9 and 10 on dynamics come to mind.

http://www.feynmanlectures.caltech.edu/I_toc.html

 

[zahbaz]

Also, a lot of formulas do not have integer exponents. Radicals are fairly common. But what would you rather read:

$$K = \frac{1}{2}mv^2$$ or $$v = \sqrt{\frac{2K}{m}}$$ ?

Perhaps the refined question would be, "why not irrational exponents?"

 

[Ken G]

I mean why is f=ma?
why not m^0.123a^1.43 or some random non integers?

Here's my take on that-- we look to define concepts that work simply, and when we succeed, it ends up meaning that the formulae have simple exponents.

Look at the F = ma example. Notice that we could just as easily define "Lorce" as e raised to what we now call force. To fix the units, we would have some reference value of "Lorce = e", and that would correspond to some reference force, where Lorce = e^{(force / reference force)} . Now we get the exact same physics, all the same predictions to every observation, using the law:
(ma / reference m * reference a) = ln(Lorce)
That doesn't look anything like an integer power, but it's the same law. An important thing to note is how "Lorces" would work. When you have two lorces on the same body, you don't add them, you multiply them. All the physics is the same, every prediction just as accurate.

But if we really did have that law, what would happen? It would not take long for someone to notice that instead of defining this "Lorce" concept, it would make more sense to define the "force" concept by taking the natural log of the Lorce, because that new concept would be additive-- the force on a body is then the sum of all the forces on its parts. We look for concepts that have nice properties, and the formulas they engender will reflect those nice properties. Why there is anything with "nice properties" is a very deep and difficult question-- why does the universe make sense at all to intelligent apes? Either it really does have some kind of mathematical design that our minds are programmed to understand somehow, or perhaps we merely focus our attention on the things that we have noticed we are able to understand, choose idealizations and simplifications that connect with what we can understand, and lo and behold, we find that the formulas we get are understandable.

 

[Dadface]

I think in many cases the integer exponents occur in basic simplified forms of the equation.

Take for example Ohms law: V=IR. This simple equation applies only if R is constant, but R depends on factors such as temperature.

We could write a more detailed equation which takes into account factors such a linear dimensions,resistivity and temperature coefficient of resistance but that equation would itself be simplified because, for example, the temperature of the sample can vary with the temperature of the surroundings. Amongst other things a temperature change in the surroundings will introduce a time varying exponent into the equation.

 

[sophiecentaur]

People seem to be missing part of the point here. By choosing a small enough range of your variables, you can make a formula with integer exponents - or even straight line relationships, to describe any relationship. But that formula will apply only over that range. No simple formula works over an unlimited range of variable values. There is no magic involved - it's just pragmatism. Imo, the OP is putting the (integer) cart before the (real) horse.

 

[Demystifier]

I mean why is f=ma?
why not m^0.123a^1.43 or some random non integers?
I hope you understand that my doubt doesn't limit just to force or energy or velocity e.t.c.
it also extends to area of a square,circle e.t.c and all other formulae
i think whole thing starts with direct proportionality.

This is actually a good question, and I think I have a good answer.
In fact, most mathematical relations between physical quantities can not be written by simple equations involving only integer powers. Most realistic systems in nature are non-linear or complicated by other means, so that we cannot write down a complete mathematical expression that describes them. It's only a relatively small subset of all phenomena that can actually be described by simple equations, and only in that cases we actually write down the equations. The result is that most written equations are simple, despite the fact that most equations (written or not) are not simple. In other words, we tend to explicitly write down equations only when they are simple, which, among other things, means - only when their exponents are simple. And of course, integer exponents are simpler that non-integer ones, so we tend to write down equations only when the exponents are integer.

 

[[Quadratic]]

F, m, a, L, etc are just symbols we use to represent concepts. If m=2 then m^0.123=2^0.123.

If you are asking why the equations that describe the universe are not strictly linear then it's an interesting question but not one I'm certain can be answered.

 

[russ_watters]

Let me repeat: most equations have to be integers due to the requirements of mathematical logic and geometry. F=m^1.2 * a (for example) isn't logically possible:

Consider f=ma a little more closely with this example:

Say, you have two 1kg weights and you apply a 1N force to each of them. They each accelerate at 1m/s^2.

Now you tie them together with a string and repeat. Why should the acceleration be different if they are connected than if they weren't?

 

[sophiecentaur]

Let me repeat: most equations have to be integers due to the requirements of mathematical logic and geometry. F=m^1.2 * a (for example) isn't logically possible:

You can't be sure that statement would apply near a black hole, for instance and it definitely wouldn't apply 'near' the quantum level. All formulae are simplifications and the extreme simplification is a linear relationship. Next simplest involves integer exponents. etc. etc.
You don't have to go far to find non integer exponents. Take the voltage decay in an RC discharge.

 

[Ken G]

Let me repeat: most equations have to be integers due to the requirements of mathematical logic and geometry. F=m^1.2 * a (for example) isn't logically possible:

Logic does not rule out an expression like that, without implicit assumptions. Let's go back to your example of two masses falling together, with a string between them. I can easily define the "mash" of those objects to be what you would call mass^1/1.2, and if I use the letter "m" to denote "mash", then I do get F = m^1.2 a. So there is nothing logically impossible about that formula. What you are saying is that the "mash" won't have the property that the mash of a system is the sum of the mash of all its parts, but so what? Who said that must be true-- as sophiecentaur points out, that's actually not true for mass in high-gravity systems, and it's not true for the quarks inside nucleons. Still, we can agree that we are never going to define "mash" this way, if we have a perfectly working concept of "mass" that does, at weak gravity levels and outside nucleons, work like the mass of the whole is the sum of the masses of the parts. So that's what I mean that we are the source of our own simple equations-- we build the equations around concepts that obey simple rules, like the whole is the sum of the parts, and if we make our building blocks have simple properties, we end up with simple equations that invoke those building blocks.

 

[russ_watters]

Logic does not rule out an expression like that, without implicit assumptions. Let's go back to your example of two masses falling together, with a string between them. I can easily define the "mash" of those objects to be what you would call mass^1/1.2, and if I use the letter "m" to denote "mash", then I do get F = m^1.2 a. So there is nothing logically impossible about that formula.

When you try to apply that to reality, it creates a logical contradiction: stringing a slack rope between the two objects would cause their acceleration to increase by 44% without any identifiable cause.

 

[Khashishi]

Exponentiation is essentially repeated multiplication. Of course, we know how to define non-integer exponents mathematically, but these are less natural (farther removed from a concrete description of nature). Maybe that answer sounds like begging the question, so let me try a little more. The universe just happens to be somewhat simple. Not so simple that we have it all figured out, but simple enough that physics is possible. We can "construct" squares and cubes via a small number of mathematical steps (repeated multiplication). Starting from the basic axioms of set theory, it is simpler to construct whole numbers than integers, integers than rationals, rationals than real numbers, etc. From this simplicity argument (which has its roots in the same place as Occham's razor), we expect integers to appear in nature much more often rational numbers and real numbers.

There's another reason for our simple laws. Physics is about finding simple laws to describe nature. All physical laws are just approximations. Often, these approximations are valid in some limit where a parameter is very small or large. In these limits, we often find simple power law models even when the underlying reality is more complex. We can take a Taylor series or other power series expansion of a complex law to give simple laws expressible in terms with integer powers. Or, we might discover the integer power law through experiment and only later discover that the law is more complex when we get out of a certain limit. For example, kinetic energy
$T=\frac{1}{2} m v^2$
is just an approximation for
$T = (m^2 c^4 + p^2 c^2)^{1/2} - m c^2$

 

[Ken G]

When you try to apply that to reality, it creates a logical contradiction: stringing a slack rope between the two objects would cause their acceleration to increase by 44% without any identifiable cause.

That is not the case, you are assuming that the mash of the whole is the sum of the mash of the parts, but the defining rule of "mash" is that it is everything you would call mass, but raised to the power 1/1.2. This definition would correctly predict all experiments, using that formula, but the mash of the whole is not the mash of the parts. That is what I mean by implicit assumptions that go beyond logic.

 

[russ_watters]

That is not the case, you are assuming that the mash of the whole is the sum of the mash of the parts, but the defining rule of "mash" is that it is everything you would call mass, but raised to the power 1/1.2. This definition would correctly predict all experiments, using that formula, but the mash of the whole is not the mash of the parts. That is what I mean by implicit assumptions that go beyond logic.

I understand what you are describing and understand that it would work if we lived in such a universe. What I'm saying is that such a universe would not be logical: 1+1=2.88

 

[Ken G]

I understand what you are describing and understand that it would work if we lived in such a universe. What I'm saying is that such a universe would not be logical: 1+1=2.88

But things like that can happen in our universe, as in the examples I gave-- the mass of three quarks in a nucleon, the mass of two objects orbiting in very strong gravity. So these are not rules of logic, they involve additional assumptions to be able to get things like 1+1=2. Use different assumptions, and we can have 1+1=2.88. So this is not unlike what you are saying, but it isn't just logic-- it's also our desire to find simplifying building blocks, building blocks that obey what you are calling "logic", but it's more like "simple rules that can be axiomatized." When we look for simple rules that can be made part of a simple mathematical system, like mass instead of mash, we end up with simple equations that govern those building blocks. We get out what we put in, but other possibilities can occur in situations when our simple building blocks go outside the domain where they work so simply.

In other words, when you see that the mass of two weights is the sum of the masses of each weight, you are calling that "logical", but I am saying that is the reason we talk about the mass rather than the mash of those objects. The real question is, why is the whole the sum of its parts, and when is that not true? That will tell us when to expect simple equations like F=ma, and when to expect those simple equations to break down.

 

[russ_watters]

You can't be sure that statement would apply near a black hole, for instance and it definitely wouldn't apply 'near' the quantum level.

Im very thin on QM, but I'm pretty sure 2 apples still weigh twice as much as 1 apple, even near a black hole.

All formulae are simplifications and the extreme simplification is a linear relationship.

That's getting closer to philosophy, but I don't agree. I don't think scientists would be doing science if they didn't think they had a chance of finding the actual laws of the universe...whether they are linear or not.

Next simplest involves integer exponents.

That makes no sense. The work and kinetic energy equations are linear and quadratic, respectively, but that doesn't make either a simplification.

You don't have to go far to find non integer exponents. Take the voltage decay in an RC discharge.

Fair enough; I'm sure there are some. But there doesn't seem to be a lot.

 

[russ_watters]

But things like that can happen in our universe, as in the examples I gave-- the mass of three quarks in a nucleon, the mass of two objects orbiting in very strong gravity. So these are not rules of logic, they involve additional assumptions to be able to get things like 1+1=2. Use different assumptions, and we can have 1+1=2.88.

No. No scientist would conduct an experiment that shows 1+1=2.88 and not conclude that he's overlooked something. I don't want to tun this into an endless string of "debunk my next example" games, but In your quarks example, you improperly ignored the binding energy.

In either case, this game will take us further and further off track, while still proving my point: the majority of the basic rules of the universe - such as f=ma - are integer exponents because they follow logic.

 

[Ken G]

No. No scientist would conduct an experiment that shows 1+1=2.88 and not conclude that he's overlooked something. I don't want to tun this into an endless string of "debunk my next example" games, but In your quarks example, you improperly ignored the binding energy.

Of course I did not ignore the binding energy, the binding energy is my entire point. When you attach a string between two objects, how does "logic" tell you that you can ignore the binding energy there? It's not an issue of logic, it's an issue of experience, and choosing concepts, like mass and force, that we already know have simple properties. That's why the formulas that involve them come out simple, but when we leave the arena where those simple properties were deduced, we lose the simple properties, and the equations get more complicated. What is amazing is how wide the domains are before we encounter the complexities-- that I have no idea about, it could allow the OP to be rephrased "how is it that we have access to conceptual building blocks powerful enough to be able to generate simple theories that work?" This is a great mystery indeed.

In either case, this game will take us further and further off track, while still proving my point: the majority of the basic rules of the universe - such as f=ma - are integer exponents because they follow logic.

No, they follow simple rules because we choose as our building blocks the kinds of things that we already know follow simple rules. It is logical that we should use our experience to do that, but the result cannot be shown from logic, it requires scientific observation.

 

[russ_watters]

Of course I did not ignore the binding energy, the binding energy is my entire point. When you attach a string between two objects, how does "logic" tell you that you can ignore the binding energy there? It's not an issue of logic, it's an issue of experience, and choosing concepts, like mass and force, that we already know have simple properties.

Yes. So we apparently agree: most equations in physics contain integer exponents because the universe follows simple mathematical logic.

No, they follow simple rules because we choose as our building blocks the kinds of things that we already know follow simple rules.

Humans did not invent the universe: we do not have such power as you are suggesting.

Or from the opposite direction: we didn't choose to investigate what was simple, we chose to investigate what was there.

 

[Ken G]

You seemed to be saying that it had to hold that F=ma based on some kind of a priori logic. I'm saying we need experience to have any idea if that law is going to work, because we do not know that F or m are going to be useful and simple concepts without that experience. If all you are saying is that we have found by experience that the universe tends to obey some simply unifying principles that are conducive to the use of logic, then I agree. But it is the experiences that tell us this works, not logic-- experience tells us what we should want our force concept to be, and what we want our mass concept to be, and that experience means we use things in equations that we have found, by experience (not logic), to work out simply. It is logical to want things to come out simply, and build concepts based on that desire, but it is not logic that let's us assert we must have F = ma. Indeed, we know this equation is not generally true, because there are applications, some that I mentioned, where the mass concept does not have its simple meaning of the sum of the parts. Had our experience been in environments like that, we might not even use the concept of mass at all, so there would be no lack of proportionality in F = ma to worry about in the first place. If all this was pure logic, we would not need experience to tell us whether mass was, or was not, a useful notion.

Indeed, on further thought, it seems to me that the best answer as to why the equations come out simple is that we live in a universe that has nearly unbroken symmetries. It is likely that all these simple integer exponents can be traced to symmetries of various kinds, so because we look for symmetries and find them, we discover useful concepts like mass and force, that can be used in simple formulas. But it is not logical that the universe is close to having unbroken symmetries, that is pure experience. Without experience, we cannot know there will be any useful concepts that can be used to generate simple formulae, and no logic would help us change that.

 

[russ_watters]

You seemed to be saying that it had to hold that F=ma based on some kind of a priori logic.

"Had to" isn't essential/relevant here -- though I suspect it is true. The universe is what it is and whether it happened because it could have happened no other way, or just happened by random chance or just happened because God felt like making it this way is a secondary question to the OP's question. Whatever the reason, the universe works that way.

I'm saying we need experience to have any idea if that law is going to work...

Of course. That doesn't have anything to do with the OP's question though.

If all you are saying is that we have found by experience that the universe tends to obey some simply unifying principles that are conducive to the use of logic, then I agree.

Yep. That's all the OP's question is about and all my answer says.

 

[sophiecentaur]

Of course, we know how to define non-integer exponents mathematically, but these are less natural (farther removed from a concrete description of nature).

Are you saying that Exponential decay and growth are not amongst the most natural bits of Nature'?

Yes. So we apparently agree: most equations in physics contain integer exponents because the universe follows simple mathematical logic

Why should"logic" come into this? Integer exponents are just seen to be there in the most simple models we use to characterise what we observe. You seem to be suggesting some God - devised set of rules that are based on elementary algebra. That's along the lines of the view of Physics that obtained in the late 19th Century. Modern Physics goes well beyond that.

 

[kayan]

"Had to" isn't essential/relevant here -- though I suspect it is true. The universe is what it is and whether it happened because it could have happened no other way, or just happened by random chance or just happened because God felt like making it this way is a secondary question to the OP's question. Whatever the reason, the universe works that way.

Of course. That doesn't have anything to do with the OP's question though.

Yep. That's all the OP's question is about and all my answer says.

The OP has asked a pretty interesting question, yet many people here fail to see the depth of the question and think the answer is all so obvious, but its not. That is the response I would get from a freshmen in college, not an inquisitive scientist. Ken is correct in everything he has said.

A very common mistake that many people on this thread as well as many scientists make, is that "nature does math". In other words, people actually think that when a particle is moving, the particle has a brain and is "smart enough" or "thinks about" the path of least action or minimum energy and then decides to take that. Then, if we study this particle long enough, we too can elucidate what "math and equations" the particle was using and call that a law of physics. This is simply not true.

Nature does what nature does (for reasons that we may never know), and we as scientists try to control/predict these behaviors using math and physics. Hence, getting back to what Ken said, the good scientist will build a model or a mental framework with basic postulates and assumptions, whereby, if we can model nature by these basic principles, then we can predict and understand many other things. However, all models will make some approximations, which means that if we find an instance in nature that violates the postulates, then the model will no longer be accurate. Russ, you keep eluding to the fact that 1+1=2. I don't know how any well grounded scientist can state this as there are many examples of when this is false (not by the laws of math/logic, but by the laws of observation/physics). Like if you have a spaceship traveling near speed of light and it turns on its headlights, a bystander would not see light going twice the speed of light, he would only see light at its normal speed.

In the end, how much do we really understand about nature. I always like the quote from Richard Feynman where he admits that even he still doesn't truly understand what the concept of internal energy is...which is a central concept of much of physics and thermodynamics that many young scientists would argue that they think they know all about.

 

[kayan]

There are some posters here that have offered interesting opinions why it is that we have the equations that we do. As I stand right now, I believe it is a mix of a lot of things, like that is how we defined the law to be and that there are many nasty nonlinear laws that are much harder to elucidate or write down, so we don't.

This question often is important in the field of transport and fluid mechanics. Often, people here study laws by how the behavior "scales" with a certain variable of interest. For example, if a law is a power law with a certain non integer exponent, that could mean the system is dominated by diffusion or convection, or perhaps by consumption from a chemical reaction, etc... So in addition to the OP, there is at least this field of physics that is very interested in how behaviors in nature scale with certain variables and why.

 

[QuantumCurt]

I think this is a really good question, but I think it's also more accurately considered in terms of two questions. One can ask this question in a purely mathematical sense, or one can ask this question in a philosophical sense.

Why is F=ma? Simply put...because we said so. It's important to understand how these quantities arise. Someone did not 'discover' force one day and decide to mathematically quantify it. It's really quite the opposite. The product of the mass of an object and the acceleration of the object happens to be a useful quantity. What is acceleration? What is velocity? It is the rate of change of position with respect to time. The simplest and most logical way to define our velocity is the unit of $\frac{1~meter}{1~second}$, or simply $\frac{m}{s}$. We could write it as $\frac{m^{1/3}}{s^{1/3}}$ if we wanted to. Now our velocity is defined in terms of 1 cube root meter per 1 cube root second. This is still a useful unit. It gives us a rate of change of position with respect to time, and it uses defined quantities. However, is this a logical quantity? We can define our base units as whatever we'd like to define them as. What if we want to figure out how quickly the velocity is changing? We want to find the rate of change of velocity with respect to time. Since velocity is 1 cube root meter per 1 cube root second, the rate of change of velocity with respect to time is logically defined as 1 cube root meter per 1 cube root second per 1 cube root second. $\frac{\frac{m^{1/3}}{s^{1/3}}}{s^{1/3}}$, which we can then write as $\frac{m^{1/3}}{s^{2/3}}$.

Now, we want to measure the acceleration of an object in free fall. Using our base unit of $\frac{m^{1/3}}{s^{2/3}}$, we find a value of $g=2.13997\frac{m^{1/3}}{s^{2/3}}$. This is a very useful quantity...but there's one problem. Writing the units out gets tiresome, and it doesn't look like a nice, clean unit. What do we do? We cube both sides of the equation.

$$g^3=(2.13997\frac{m^{1/3}}{s^{2/3}})^3$$
$$g^3=(2.13997)^3\frac{m}{s^2}$$
$$g^3=9.80\frac{m}{s^2}$$

This is obviously a very familiar number. It's the force of gravity that we all know. But now it's equal to $g^3$. This isn't a problem though. We were using the cube root meter and the cube root second as our base units before, but we simply decided to redefine our base quantities. We don't have to call it $g^3$...we can simply call it $g$.

Point being, we define our units. If we further defined the cube root kilogram and multiplied it by the "cube root acceleration", we would get a quantity

$$ma=kg^{1/3}\frac{m^{1/3}}{s^{2/3}}$$

We can redefine each of these units as 1 cube root meter = 1 meter etc., and derive a quantity of

$$ma=kg\frac{m}{s^2}$$

As it turns out, this is a useful quantity. Useful enough that we turn it into one of the main ideas in physics. Instead of defining it as the product of 1 kilogram and 1 meter per second per second, we call it a force, and define our unit of force as the Newton, and write a force in terms of N, rather than the complicated units. We can choose any units we want to choose. We could choose a force unit of 40 kilograms times 6.5 meters per 4 seconds per 4 seconds if we wanted...but we could simply redefine the value of 1 kilogram as being equivalent to the quantity of 40 kilograms as it's defined today, and do the same with the other values. Since we get to choose the base units that define our measured quantities, it makes sense to choose the simplest ones.