Fundamental laws and differential equations

[dirad]

why are many fundamental laws of nature formulated in the form of differential equations?

 

[Simon Bridge]

Because that is convenient for working out more specific solutions.

 

[dirad]

why are they used in the first place? what makes them so general? why are derivatives of various orders included in the description of nature?

 

[SteamKing]

Because not everything in nature is static. Some things depend on the rate of change of other things, to be hand-wavingly vague about it.

 

[dirad]

how do we measure a rate of change experimentally?

 

[Adyssa]

If you measure something, or some group of things, every day for a year, then you have a bunch of data. You can plot the data on a graph, measurement vs time, and draw a curve through the points, or rather, find a function that determines the curve. Then you can "do calculus" and determine the rate of change at any point on the curve. So you might measure the rate of change of the number of frogs in your front yard. Or the rate of change of temperature, or clouds, or, if you're patient, trees. Calculus just happens to be the language we use. It gets complicated when things depend on many other things.

 

[dirad]

If you measure something, or some group of things, every day for a year, then you have a bunch of data. You can plot the data on a graph, measurement vs time, and draw a curve through the points, or rather, find a function that determines the curve. Then you can "do calculus" and determine the rate of change at any point on the curve. So you might measure the rate of change of the number of frogs in your front yard. Or the rate of change of temperature, or clouds, or, if you're patient, trees. Calculus just happens to be the language we use. It gets complicated when things depend on many other things.

if i plot data on a graph and draw a curve, then i have the required variables. so, why calculate a rate of change?

 

[DeIdeal]

if i plot data on a graph and draw a curve, then i have the required variables. so, why calculate a rate of change?

Well, because there are things that depend on the rate of change

Things which we usually model with differential equations! We can do measurements to confirm and/or come up with these models, and then use these models to predict behaviour of different systems.

 

[jackmell]

why are many fundamental laws of nature formulated in the form of differential equations?

In particular, second-order differential equations. But laws we design describe phenomena in Nature and Nature is a changing phenomenon, our Universe is a dynamic (changing) system. Planets are revolving around stars, stars change, life changes. Differential equations is a math of change: the derivative is a change-descriptor, it describes change. So not surprisingly, creating laws to describe Nature would work well in the language of Differential Equations.

 

[dirad]

Well, because there are things that depend on the rate of change

Things which we usually model with differential equations! We can do measurements to confirm and/or come up with these models, and then use these models to predict behaviour of different systems.

then, i am modelling with differential equations. these equations have solutions where every thing is straightened out and we have no rates of change but we have the total change. why do we not use these solutions for modelling systems?

 

[dirad]

In particular, second-order differential equations. But laws we design describe phenomena in Nature and Nature is a changing phenomenon, our Universe is a dynamic (changing) system. Planets are revolving around stars, stars change, life changes. Differential equations is a math of change: the derivative is a change-descriptor, it describes change. So not surprisingly, creating laws to describe Nature would work well in the language of Differential Equations.

the derivative is a change-descriptor, but so are total differences. derivatives appear in fundamental equations. when solved, total differences appear. why are the latter not used in the description?

 

[SteamKing]

Not all differential equations have nice and tidy solutions. Many differential equations can only be solved by numerical means. Analyzing the gravitational interactions is a system containing three or more bodies is one type of problem which can only be analyzed numerically in general, although there are some special cases which have analytical solutions.

 

[DeIdeal]

then, i am modelling with differential equations. these equations have solutions where every thing is straightened out and we have no rates of change but we have the total change. why do we not use these solutions for modelling systems?

Could you post an example of what you mean? If I get what you mean, solutions like that usually only work in very specific situations, whereas the model works in a more general one.

 

[dirad]

Could you post an example of what you mean? If I get what you mean, solutions like that usually only work in very specific situations, whereas the model works in a more general one.

s=1/2 at² where s= displacement of falling body, t= total time, a= acceleration due to Earth gravity

 

[DeIdeal]

s=1/2 at² where s= displacement of falling body, t= total time, a= acceleration due to Earth gravity

Good, that's what I guessed you we're talking about. It's, as I assume you already know, derived from Newton's m\ddot{r}=F. But, and this is the big thing, it only works when F, the force, is constant (for free fall F=-g~9.8 m/s²), and the initial velocity of the object is zero.

Now, what would you do if F=-kr and the system is the (in)famous harmonic oscillator? The differential equation we use to model the system still works (as long as we're satisfied with the classical approximation), but the solution is very different. This is why the models are often more important than their specific solutions, and the models are often differential equations.

 

[dirad]

Good, that's what I guessed you we're talking about. It's, as I assume you already know, derived from Newton's m\ddot{r}=F. But, and this is the big thing, it only works when F, the force, is constant (for free fall F=-g~9.8 m/s²), and the initial velocity of the object is zero.

Now, what would you do if F=-kr and the system is the (in)famous harmonic oscillator? The differential equation we use to model the system still works (as long as we're satisfied with the classical approximation), but the solution is very different. This is why the models are often more important than their specific solutions, and the models are often differential equations.

thanks. this gets us back to first question. basic laws like Newton's 2nd law are differential equations containing derivatives. the same goes for maxwell's ,schroedinger's,einstein's equations. is there some general solution to each of these equations that can be considered as fundamental law in its own right?

 

[DeIdeal]

thanks. this gets us back to first question. basic laws like Newton's 2nd law are differential equations containing derivatives. the same goes for maxwell's ,schroedinger's,einstein's equations. is there some general solution to each of these equations that can be considered as fundamental law in its own right?

I'm not sure if what you wrote is literally what you meant, but no. The equations do not describe the same phenomena, so there's no reason why they would have the same solution (that said, this doesn't mean that equations describing different phenomena can't have the same solution, just that they don't have to). Besides, they are all differential equations, but they are not the same equations. You wouldn't expect all polynomial equations to have the same, explicit solution, would you?

EDIT: Wait, I might've misread that. The answer is still no, though. For example, in Newton's law, the form of the force is unspecified, which essentially means that there's no general solution to the problem. Take the N-body system mentioned above as an example of a system where we can't necessarily even construct a solution.

Now, as to why the "basic laws" contain derivatives, I'll quote SteamKing's post:

Because not everything in nature is static. Some things depend on the rate of change of other things, to be hand-wavingly vague about it.

There are reasons for some particular differential equations being especially common in nature, but they are somewhat involved.

 

[dirad]

using diff.eqs.contain some hidden assumptions. we assume continuity and smoothness of variables, for example. we associate simultaneous cause (force) and effect (acceleration). why do we almost automatically resort to them ?

i have read some accounts on the role of calculus in mathematics in which integration and differentiation are considered as extensions to usual mathematical operations of addition, subtraction,multiplication, division, raising to a power,taking of a logarithm.

 

[DeIdeal]

using diff.eqs.contain some hidden assumptions. we assume continuity and smoothness of variables, for example. we associate simultaneous cause (force) and effect (acceleration). why do we almost automatically resort to them ?

Not everything has to be assumed. In relativistic calculations, we obviously can't accept solutions that allow for instantant (or superluminal) transmission of information. For the classical heat equation, for example, that assumption is often good enough.

Assumptions are needed and used because they make calculations feasible, but we don't want to oversimplify things, either. It's a kind of situational awareness: you have to know what assumptions are reasonable for specific systems and problems.

The requirement for using r(t)=r_0+v_{0}t+\frac{1}{2}at^2 is that the force is constant (along with the requirements of using m\ddot{r}=F), the requirement for using m\ddot{r}=F requires the mass to be constant (which I neglected to mention earlier) and the system to be 'classical'. It allows for a much, much wider range of solutions, but is still possible to calculate, at least numerically, even for complicated systems, whereas solving the Schrödinger equation for the same systems is simply out of the question, even though we know for a fact that it's more accurate (while still not relativistic, for example!).

Methods of solving differential equations are well known, both numeric and analytic. The "requirements" for using differential equations in general to model things are diminutive when compared to the advantages they offer.

EDIT: A couple of edits and additions here and there.

 

[Mandelbroth]

Why are many fundamental laws of nature formulated in the form of differential equations?

Because sometimes quantitative things change and their changes are often parametrized by $n$ variables. For some reason, everyone wants to find out about things that change.

I like to generalize.

s=1/2 at² where s= displacement of falling body, t= total time, a= acceleration due to Earth gravity

But...why? Is this always true?

Consider that gravitational acceleration, $g$, changes with altitude. Thus, we should really say that, assuming a generic "up" is the positive direction, $g=g(r)=-G\frac{m_{\text{Earth}}}{r^2}$, where $r$ is the distance from the Earth's center of mass, $m_{\text{Earth}}$ is the mass of the Earth, and $G$ is the gravitational constant. Thus, gravitational acceleration is not constant with changing altitude, and $\frac{a}{2}t^2$ is not necessarily a good approximation for $s$.

As a more general example, say we want to know an object's linear position over time. Typically, we approximate $x\approx x_0+v_0t+\frac{a}{2}t^2$, but this only works if $a$ is constant. People who know a sufficient amount of calculus will immediately recognize the formula as a truncated power series. In fact, if we assume that the position function $x(t)$ is infinitely differentiable, we get $\displaystyle x(t)=\sum_{n=0}^{\infty}\left[\left.\frac{d^n x}{dt^n}\right|_0\left(\frac{t^n}{n!}\right)\right]$, where $\frac{d^nx}{dt^n}|_0$ is the $n^{\text{th}}$ derivative of $x$ at 0. Note that the above power series works even when $a=\frac{d^2x}{dt^2}$ is constant. We can then truncate the power series to obtain an arbitrary degree of accuracy. That's part of the beauty of differential equations: we can make accurate and helpful mathematical deductions for use in science. The world is naturally full of change, and thus differential equations provide an excellent and often smooth way of describing natural phenomena.

 

[WannabeNewton]

why are many fundamental laws of nature formulated in the form of differential equations?

Because many of the quantities in physics are defined in terms of various kinds of derivatives. There's really nothing deep in this. Calculus is simply necessarily to describe said physical quantities, algebra isn't enough.

 

[Simon Bridge]

In Nature, some things depend on the way other things change. i.e. speed depends on how position changes with time.

When we see relationships that depend on continuous changes in nature, then we have to describe that somehow.

The language we use is mathematics and calculus happens to be a useful way to describe how changing things depend on each other. We have to describe it somehow after all - so why not this way?

How else could it be done?

I wonder if this is not more properly discussed under "philosophy"?