Differential equations in physics

Good question :D

 

I can only second that !

Of course, there is an answer, but it is just begging the question. The answer is that spacetime has symmetries of translation and uniform motion. This makes you pick "positions" and "velocities" as initial conditions. But as the question *why* does spacetime have these symmetries, the answer is then that these leave the laws of nature invariant, which are... second order diff. equations...

 

Why do the laws of nature have to be second order diff. equations in order to be invariant ?

 

No, it is the other way around (hence why this is begging the OP's question). BECAUSE we find that the laws of nature are second-order, we NOTICE that they are invariant under certain transformations, and hence we call these invariance laws, the invariance of spacetime.

If you look at Newton's law, F = m a, you see that it is invariant under translations : x' = x + u, and you see that it is invariant under uniform motion: x' = x + v t. This means that you should be able to choose a "u" and a "v", two initial conditions. Hence (at least) second order. However, Newton's equation is not invariant under uniform acceleration: x' = x + g t^2 is not an invariance of Newton's laws. So no third order.

So what came first ? Newton's law, from which we deduced invariance under translation and uniform motion ? Or invariance under translation and uniform motion, from which we knew that we would have a second-order differential equation ?

 

Somewhat related to the OP, in the case of motion, most physics related problems involve forces or accelerations, and in some cases (like aerodynamic drag), the equations for those forces or accelerations are too complicated to be integerated directly, so we're left with 2nd order differential equations. There is a 3rd order effect, jerk, which is the rate of change in acceleration. One case where 3rd order effect (jerk) would be important would be a automotive simulator, since there are flexible components in an automobile, such as suspension and tires, and jerk affects how these components respond.

I'm not sure how often 3rd order effects exist but are ignored, as opposed to aspects of nature that are truly limited to 2nd order effects.

 

Another place I've see the third derivative "jerk" and maybe even higher order derivatives considered is in designing the shape of automotive cams - since the follower or valvestem motion follows (or should follow) the shape of the cam, and acceptable valve seat wear depends on letting the valve down gently.

 

But if there is a non-zero jerk over time then surely there has to be a non-zero d^4x/dt^4 and a d^5x/dt^5, etc. Why are these not considered?

 

Given a solution to Newton's equation:

x''(t) = F(x,x'), x(0) = a , x'(0) = b

It is trivial to solve for x'''(t), etc, by simply differentiating. This accounts for the cases with non-zero jerk.

First-order equations cannot oscillate, they can only grow or shrink or shrink towards a limit point. Second-order equations can oscillate, and they always either become unbounded or go into a steady state (either a fixed point or a periodic oscillation) as t -> Infinity. Equations of order 3 and higher can have chaotic oscillations.

Remember that a system of n second order equations is equivalent to a single equation of order 2n. So a third degree equation for modeling shock absorption is a way of abstracting the complex process.

In quantum mechanics the justification is much more straightforward. Here we use a second order partial differential equation because its solutions are members of an infinite-dimensional vector space. The choice is arbitrary subject to the axioms of quantum mechanics. Another alternative is to use infinite matrices.

 

This is a great question, and I've spent some time trying to come up with a decent reason. I don't have a complete answer, but I got a good hint from "Conceptual Foundations of Contemporary Relativity Theory", and here it goes:

Let's start with the basic dynamical force equations: Newton's gravitational and Maxwell's electrodynamic equations. Both of these have a form of

F = k/r^2

These equations are important becasue they relate *kinematic* things (distances, accelerations, velocities, charge, time..) to *dynamic* things (forces).

I need to stress that equations of motion are extremely fundamental things, much more fundamental than other types of equations (constitutive relations, disperison relations, etc). Things like Hamilton's equations, or Lagrange's equations, or Shrodinger's equations, Einstein relations, etc... are rooted as equations of motion, even thought they may superficially look more complicated.

Ok- F = k/r^2 is a fundamental equation in physics. It is also known that the 1/r^2 part is due to their being 3 spatial dimensions. So the equations of motion reflect a fundamental property of space.

Now here's the important part: the equation F = k/r^2 is an integral form of Poisson's equation [tex]\nabla^{2}\phi=\rho[/tex], or alternatively Laplace's equation [tex]\nabla^{2}\phi=0[/tex]. Consequently, my book makes the following assertion [pg 179] , and the part I don't fully understand:

"Any physical law must be a partial differential equation containing no derivatives higher than the second, and that the law must be linear in the second derivative."

It goes on to state that "there are no known cases where third (or higher) order differential equations are required in basic laws, nor do we have conceptual resources for interpreting them, but nonlinear equations and combinations of first and second derivatives are known to occur".

Apparently Eddington felt that this was too restrictive, and the reason is "unwarranted bias". However, Schrodinger stated that "The great acheivements of Newton's laws was to concentrate attention on the *second* derivatives- to suggest that *they*- not the first or third or fourth, not any other property of the motion- ought to be accounted for by the environment."

As I said, this is a fascinating topic, and I don't have a good answer for it. However, I think a good explanation is simply that the three-dimensional nature of space leads to physical laws being expressed as second-order differential equations. I would be most interested to hear from anyone else on this.

 

man...most of these ideas are waaay beyond me.
Ive always just thought...well Diffeq's are math, and physics is the application of math to real life.

Well that really doesnt answer it, all I know is that if I see another heat equation problem anytime soon I just might jump off a bridge.

everything in the entire worldis calculus. last year my Matlab profesor showedus some of his projects on differentiantial equations...apparently even the Cruise Control in everyone's cars is some sort of 4th order differential equation.

 

There is an equation in structural mechanics which is of 4'th order in 1D (bending of a rod with a known cross-section), but the 2'nd and 3'd derivatives have physical meaning, and could hence be used as boundary conditions. This in contrast to quasi-corrections to differential equations which gives contributions like [tex]\nabla^5[/tex] etc. where there is no physical meaning of these higher derivatives boundary conditions. There is for example no meaning of [tex]\nabla^2[/tex] as a boundary condition in the Schrödinger equation. In short words this is the problem of defining "mathematically well formed" equations. 1)Meaningful boundary conditions

2)Not many equations survive invariance under translation and rotation as been mentioned here, and those who do are the ones we know...

3) Most PDE's are derived from conservations of something physical like charge, involving incoming flux through a surrounding surface. Applying Gauss, stokes and Green's laws to transform to volume integrals leads to 2'nd order derivatives like the laplacian operator.

 

I agree there are higher-order equations in physics: the stream function is a biharmonic, The Korteweg-deVries equation, the elastic wave propogation equation, I'm sure there's many more. But none of those have been elevated to a plysical *law*. The equations above are derived from more fundamental concepts.

 

This is a very interesting topic.
Maybe there are different laws involving higher order differential equations which are invariant under some other strange symmetries that we would never consider as natural because of our specific view of the world? Maybe there are conscient beings passing through time in a different manner, perceiving a different kind of causality that doesn't affect ours, seeing a totally different world and they would never see Newtons laws or similar as something natural, and they would never imagine that there are beings like us travelling through the world in a different direction? And maybe there are more general laws in physics still to be discovered by us and by them, some laws that we might have in common. And maybe I'm just talking nonsense? Probably so...