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45 votes
Accepted

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

In molecular dynamics simulations, the overwhelming part of the computational time is spent evaluating forces. For this reason, since the very beginning of the method, the algorithms of choice were ...
GiorgioP-DoomsdayClockIsAt-90's user avatar
39 votes
Accepted

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

On the other hand, it requires only two initial conditions x(0) and x˙(0), to obtain the function x(t) by solving Newton's equation For notational simplicity, let $$x_0 = x(0)$$ $$v_0 = \dot x(0)$$...
Alfred Centauri's user avatar
34 votes
Accepted

What situations in classical physics are non-deterministic?

There are two famous cases in classical mechanics that fail to be deterministic. The first, and most famous, is Norton's Dome, which corresponds to a system with a force of the form $$F = \sqrt{r} $$...
Slereah's user avatar
  • 16.4k
30 votes
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Why can all solutions to the simple harmonic motion equation be written in terms of sines and cosines?

This follows from the uniqueness theorem for solutions of ordinary differential equations, which states that for a homogeneous linear ordinary differential equation of order $n$, there are at most $n$ ...
Emilio Pisanty's user avatar
27 votes

How can the solutions to equations of motion be unique if it seems the same state can be arrived at through different histories?

You lack complete knowledge of the system you're asking about. You only know about the jar. And if the complete system is the jar then yes, it has always been empty because there's nothing to ...
Schwern's user avatar
  • 4,534
25 votes
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Is there a general way of solving the Maxwell equations?

You need to be more precise about exactly what problem you're solving and what the inputs are. But if you're considering the general problem of what electromagnetic fields are produced by a given ...
tparker's user avatar
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25 votes
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Why can't we run the laws of physics backwards and forwards in time infinitely?

Let's not even talk about big bangs yet. Consider a simple non-linear ODE $\frac{dx}{dt}=-x^2$ with the condition $x(1)=1$. There is a unique maximal solution defined on a connected interval, which in ...
peek-a-boo's user avatar
  • 6,605
22 votes
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Solution to pendulum differential equation

The pendulum problem can be solved exactly if an elliptic integral is used. The elliptic integral in question is defined via \begin{equation} F(\phi,k)=\int_{0}^{\phi}\frac{dt}{\sqrt{1-k^{2}\sin^{2}t}}...
wong tom's user avatar
  • 567
17 votes
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What does is really mean to say that a 3-body problem is not solvable?

The three-body problem lacks a closed-form solution, which is a mathematical expression that uses a finite number of "standard" operations (addition, division, logarithm, etc.), usually ...
Nuclear Hoagie's user avatar
14 votes

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

In molecular dynamics you have to evolve a large number of particles, so efficiency is important. The Verlet algorithm provides numerical stability, as well as other properties that are important in ...
Quillo's user avatar
  • 5,048
14 votes

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

GiorgioP has the right answer, but I just want to emphasize a point. We use different time integration methods for different problems. For example, consider these two problems. You want to know the ...
WaterMolecule's user avatar
13 votes
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For a square plate heated to $T$ on only one edge, how can I show that the temperature in the centre is $T/4$?

Just looking at physics, if the initial problem has a solution and you rotate the plate of $\pi/2$ you obtain another solution with boundary conditions rotated of $\pi/2$. Perform the same procedure ...
Valter Moretti's user avatar
13 votes

How can the solutions to equations of motion be unique if it seems the same state can be arrived at through different histories?

As you have noted, to solve the differential equation describing the system you need initial conditions. The condition "jar is empty" does not have a unique solution without specifying the conditions.
jim's user avatar
  • 3,857
13 votes
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What's wrong with this "proof" that all Hamiltonians are time independent?

This was too long for a comment so I'll add it as answer. You have reformulated that the following holds along a trajectory, $ dH(p,q,t) = \frac{\partial H}{\partial p}dp +\frac{\partial H}{\partial q}...
Hans Wurst's user avatar
  • 1,544
13 votes

Is resonance a general property of second-order differential equations?

Consider the second order differential equation \begin{align} f'' - \alpha^2 f = C \cos(\omega t) \end{align} with $\alpha$ a real constant. (Note the sign of the second term, which makes this ...
d_b's user avatar
  • 8,319
11 votes
Accepted

How do Maxwell's equations uniquely determine ${\bf E}$ and ${\bf B}$ despite no. of equations exceeding no. of unknowns?

Provided that the first two equations hold true at the initial condition, they are redundant for the time evolution, because $$\nabla \cdot \frac{\partial \mathbf{E}}{\partial t} = \frac{1}{c^2} \...
knzhou's user avatar
  • 103k
11 votes

Solution to pendulum differential equation

It is not possible to express the solution of the equation in terms of elementary functions. Nonetheless, you can obtain an approximate solution via numerical integration. The figure shows the ...
Prallax's user avatar
  • 2,919
11 votes

Why is the Yukawa potential equal to Green's function for free space?

Helmholtz equation is identical with the coordinate part of Klein-Gordon equation, which is the wave equation for a massive particle, so no wonder that they produce identical solutions. Note also that ...
Roger V.'s user avatar
  • 59.4k
10 votes

Norton's dome and its equation

The dome equation is expressed in terms of arc length and height, which hides a lot of bad behaviour. As Luboš has pointed out, beyond a certain point it is no longer physical so we need to constraint ...
Gruff's user avatar
  • 209
10 votes
Accepted

Can I see separation of variables as a tensor product?

Yes, that is exactly what separation of variables is in terms of the Hilbert space - generally, we have that $L^2(X\times Y) = L^2(X)\otimes L^2(Y)$, i.e. the space of square-integrable functions on a ...
ACuriousMind's user avatar
  • 126k
10 votes

Why can't we run the laws of physics backwards and forwards in time infinitely?

Your accepted answer explains why we cannot do it today - mostly because we simply are not capable to solve the equations we came up with, and even for incredibly simple examples like ${dx}/{dt}=-x^2$ ...
AnoE's user avatar
  • 2,758
9 votes

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

Power series expansion does not hold for all functions $f(t)$ or for all $t\in\mathbb{R}$, but only for real analytic functions and for $t$ in the radius of convergence. In particular, it does not ...
yuggib's user avatar
  • 12k
9 votes
Accepted

Why are the left- and right-hand sides of a differential equation with two separated variables equal to a constant?

There are two logical options when you vary $t$: either the value of the left-hand side changes, or it doesn't. If it changes, then the right side must change as well, since they are equal. But the ...
interoception's user avatar
9 votes

What's wrong with this "proof" that all Hamiltonians are time independent?

When you divide $dp/dt$ by $dq/dt$ and apply the chain rule to get $\dfrac{dp}{dq}$, you're implicitly assuming that $p$ is a function of $q$ only, thereby assuming time independence of $\mathcal H$.
user256872's user avatar
  • 6,621
9 votes
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Separable Hamiltonian systems in quantum mechanics

I think the other answer is (at least partially) mistaken. Let me first answer your question, and then explain why I have issues with the other answer. Suppose you have a Hamiltonian of the form $H = ...
Zack's user avatar
  • 3,068
9 votes
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Kepler problem in cartesian coordinates

Absorb the dimensional GM into the units, to remove excuses for not recognizing the plane-geometry structure. Note the rotational and translational invariance to be used in fixing your coordinate ...
Cosmas Zachos's user avatar
8 votes

Complete vs General Integral of first order PDE

In the general theory of partial differential equations and specifically for First-Order Partial Differential Equations one defines the general solution(Landau's general integral) and the complete ...
Keith's user avatar
  • 728
8 votes
Accepted

How to make sense of quantum fields differential equations?

The fields of a QFT are not functions of the spatial coordinates $\boldsymbol x\in\mathbb R^n$, but operator-valued distributions (borrowing Wightman's terminology). The notion of the fields being ...
AccidentalFourierTransform's user avatar

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