Interesting differential equation

In summary, the conversation discusses an equation for comparing the average speed of a group of points with their individual speeds, involving velocity vectors and a constant factor. The equation can be rewritten in different forms and there are different approaches to solving it, such as taking the square root or considering it as a path in a Lie group.
  • #1
birulami
155
0
Comparing the average speed of a bunch of points with the individual points' speeds, I came across the following equation:

[itex]\left(\frac{dx(t)}{dt}\right)^2
= \frac{1}{N^2} c^2 \sum_{i\neq j} \left(1-\frac{v_i(t) v_j(t)}{c^2}\right) [/itex]

where the [itex]v_i[/itex] are the velocity vectors (3 dimensions) of the N points. They fulfil the equation [itex]|v_i(t)|^2 = c^2[/itex]. If I didn't loose some constant factor, the equation above should be the same as

[itex]\left(\frac{dx(t)}{dt}\right)^2
= \frac{1}{N^2} \sum_{i< j} (v_i - v_j)^2[/itex]

Any chance to solve one or the other for [itex]x(t)[/itex]? I hesitate to take the square root and try to integrate the square root of the sum. Are there better ways to solve this?

Harald.
 
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  • #2
There is a mistake in the second formula which should be ##\left( \frac{dx(t)}{dt}\right)^2 = \frac{1-c^2}{N^2} \sum_{i< j} (v_i - v_j)^2##. It can be rewritten as
$$
\dfrac{N^2}{1-c^2}\, \ddot{x}(t)^2 = \left|\left|\begin{pmatrix}v_1\\v_2\\v_3\end{pmatrix}\times \begin{pmatrix}1\\1\\1\end{pmatrix}\right|\right|^2 =\left|\left|\begin{pmatrix}0&-1&1\\1&0&-1\\-1&1&0\end{pmatrix}.\mathbf{v}\right|\right|^2
$$
So the entire equation looks a bit like ##\gamma \,||\dot{\mathbf{v}}||^2 = ||\mathbf{A.v}||^2## or ##\mathbf{\dot v} = \dfrac{\sqrt{1-c^2}}{N}\mathbf{A.v}## with an exponential function as solution.

This leads to another idea, namely to consider ##x(t)## as path in a Lie group, presumably ##\operatorname{SO}(3)##.
 

Related to Interesting differential equation

What is a differential equation?

A differential equation is a mathematical equation that relates a function with its derivatives. It is commonly used to describe the change of a physical quantity over time.

What makes a differential equation interesting?

A differential equation is considered interesting if it has unique solutions or if it can be solved using different methods. It can also be interesting if it has real-life applications in various fields such as physics, engineering, and economics.

What are the different types of differential equations?

There are several types of differential equations, such as ordinary differential equations, partial differential equations, and stochastic differential equations. Ordinary differential equations involve a single independent variable, while partial differential equations involve multiple independent variables. Stochastic differential equations take into account random fluctuations in the system.

How are differential equations used in science?

Differential equations are widely used in science to model and study various phenomena, such as population growth, chemical reactions, and fluid dynamics. They are also essential in the development of scientific theories and in predicting future behavior of systems.

What are some famous examples of interesting differential equations?

Some famous examples of interesting differential equations include the Navier-Stokes equations for fluid dynamics, the Schrödinger equation for quantum mechanics, and the Black-Scholes equation for financial markets. These equations have had a significant impact on their respective fields and have been extensively studied by scientists and mathematicians.

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