# Tag Info

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### 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 ...
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### 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 $$x_0 = x(0)$$ $$v_0 = \dot x(0)$$...
• 59.8k
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### 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 $$F = \sqrt{r}$$...
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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 $n$ ...
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### 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 ...
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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 ...
• 48.1k
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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 ...
• 6,605
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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 F(\phi,k)=\int_{0}^{\phi}\frac{dt}{\sqrt{1-k^{2}\sin^{2}t}}...
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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 ...
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### 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 ...
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### 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 ...
• 1,988
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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 ...
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### 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.
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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 ...
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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 ...