-1

We found some interesting insights in differential equations of the form

$y^{(n)}(x)+F_\lambda(y(x),y'(x),...,y^{(n-1)}(x))=0$,

i.e. for ordinary differential equations of $n$-th order with $n\geq2$. The function $F$ is polynomial which can include a set of parameters $\lambda$.

We know, that in physics usually the highest derivative is of order two(?), but we are searching for applications of this kind of differential equations for $n\geq3$ in physics, engineering, or in any other area. If you have an idea or know models or theories in which such equations occur, you input would be appreciated very much.

DanielSank
  • 25,766
DGLL
  • 1

2 Answers2

1

That's just not true. If a linear system has $n$ independent ways in which energy can be stored as states, and energy can flow between these states, then you can model the system with an nth order polynomial.

Granted some systems can be approximated by a linear 2nd order rational polynomial function, but a closer look shows higher order fits might work better. For example you might assume a spring mass system as a lumped parameter 2nd order system, but find the spring has torsional as well as bending modes, and the rigid mass perhaps not so rigid. But we tend to model the states that more matter to our application, and so we might neglect the higher order modes.

And look carefully, the real world is hardly ever linear. Nonlinear systems are more the norm.

docscience
  • 11,779
0

The Dirac equation, which is a system of four first-order equations for four components of the Dirac spinor, is generally equivalent to one fourth-order equation for one component of the spinor (http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf , published in the Journal of Mathematical Physics, https://arxiv.org/abs/1502.02351)

akhmeteli
  • 27,962