Why do most formulas in physics have integer exponents?

[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.