Physics Stack Exchange
Physics Stack Exchange

Questions tagged [differential-equations]

DO NOT USE THIS TAG just because the question contains a differential equation!

DO NOT USE THIS TAG just because the question contains a differential equation!

Almost all physical systems are governed by differential equations. Hence it is often a poor way to classify a question to use this tag.

920 questions
109
votes
12 answers

Why are differential equations for fields in physics of order two?

What is the reason for the observation that across the board fields in physics are generally governed by second order (partial) differential equations? If someone on the street would flat out ask me that question, then I'd probably mumble something…
Nikolaj-K
  • 8,873
90
votes
7 answers

Do Maxwell's Equations overdetermine the electric and magnetic fields?

Maxwell's equations specify two vector and two scalar (differential) equations. That implies 8 components in the equations. But between vector fields $\vec{E}=(E_x,E_y,E_z)$ and $\vec{B}=(B_x,B_y,B_z)$, there are only 6 unknowns. So we have 8…
Warrick
  • 9,870
55
votes
6 answers

Norton's dome and its equation

Norton's dome is the curve $$h(r) = \frac{2}{3g} r ^{3/2}.$$ Where $h$ is the height and $r$ is radial arc distance along the dome. The top of the dome is at $h = 0$. Via Norton's web. If we put a point mass on top of the dome and let it slide down…
countunique
  • 1,801
37
votes
6 answers

Motion described by $m \frac{\mathrm{d}^2 x}{\mathrm{d}t^2}=-k\frac{\mathrm{d}^{\frac12 }x}{\mathrm{d}t^{\frac12}}$

What kind of motion would a (preferably dimensionless for simplicity) body do if the force acted on it was proportional to the semi-derivative of displacement, i.e. $$m \frac{\mathrm{d}^2…
user148432
  • 515
  • 4
  • 11
35
votes
4 answers

First-order wave equation: Why is its presence not common?

The (one-dimensional) wave equation is the second-order linear partial differential equation $$\frac{\partial^2 f}{\partial x^2}=\frac{1}{v^2}\frac{\partial^2 f}{\partial t^2}\tag{second order PDE}$$ that admits as its solutions functions $f$ of the…
BMS
  • 11,733
33
votes
1 answer

What situations in classical physics are non-deterministic?

In Sean Carroll's book "The Big Picture," he states (chapter 4, page 35): Classical mechanics, the system of equations studied by Newton and Laplace, isn't perfectly deterministic. There are examples of cases where a unique outcome cannot be…
WillG
  • 3,770
33
votes
1 answer

Does the heat equation violate causality?

I've ran across the idea that, besides simply writing partial differential equations in covariant form, they need to be hyperbolic with all characteristic speeds less than the speed of light. A straightforward generalization of the equations for a…
octonion
  • 9,072
32
votes
3 answers

Why aren't Runge-Kutta methods used for molecular dynamics simulations?

One of the most used schemes for solving ordinary differential equations numerically is the fourth-order Runge-Kutta method. Why isn't it used to integrate the equation of motion of particles in molecular dynamics?
WedgeAntilles
  • 481
27
votes
5 answers

Newton's law requires two initial conditions while the Taylor series requires infinite!

From Taylor's theorem, we know that a function of time $x(t)$ can be constructed at any time $t>0$ as $$x(t)=x(0)+\dot{x}(0)t+\ddot{x}(0)\frac{t^2}{2!}+\dddot{x}(0)\frac{t^3}{3!}+...\tag{1}$$ by knowing an infinite number of initial conditions…
SRS
  • 27,790
  • 13
  • 115
  • 365
27
votes
2 answers

Can one classify partial differential equations according to the causality properties of their solutions (and if yes, then how)?

Recently, I bumped into this interesting comment by Valter Moretti which made me wonder about the following, more general question (to which I suspect the answer is affirmative): Can we easily tell, just from the operators appearing in a…
Danu
  • 16,576
  • 10
  • 71
  • 110
18
votes
5 answers

Conservation of Mathematical Constraints when deriving Energy and Momentum from $F=ma$

Background: Starting from $F = ma$, integrating with respect to time, and using basic calc, one can derive $\int Fdt = m (v_f - v_i)$ Starting from $F = ma$, integrating with respect to distance, and substituting $a\ ds = v\ dv$ (from calculus),…
user1476176
  • 1,511
18
votes
6 answers

Any physical example of an "explosive" differential equation $ y' = ky^2$?

I was told that in physics (and in chemistry as well) there are processes that may be described by a differential equation of the form $$ y' = ky^2. $$ That is, the variation of a variable depends from the number of pairs of the elements. I…
mau
  • 290
18
votes
4 answers

What is the physical meaning/concept behind Legendre polynomials?

In mathematical physics and other textbooks we find the Legendre polynomials are solutions of Legendre's differential equations. But I didn't understand where we encounter Legendre's differential equations (physical example). What is the basic…
albedo
  • 1,663
16
votes
4 answers

Solution to pendulum differential equation

In a chapter on oscillations in a physics book, the differential equation $$\ddot{\theta}=-\frac{g}{L}\sin(\theta)$$ is found and solved using the small-angle-approximation $$\sin(\theta)\approx\theta$$ for small values of $\theta$, which yields the…
John Doe
  • 307
  • 1
  • 2
  • 8
16
votes
7 answers

Applications of delay differential equations

Being interested in the mathematical theory, I was wondering if there are up-to-date, nontrivial models/theories where delay differential equations play a role (PDE-s, or more general functional differential equations). It is clear that in…
András Bátkai
  • 733
1
2 3
…
61 62