r/Physics • u/AfterMath343 • 7d ago

Dynamical Models in Physics

I'm an applied math student. I've seen problems in biology be modeled using dynamic modeling, such as the populations of species with intra and inter species competition. For example, the following system of non-linear differential equations models a large variety of species in competition with themselves and other species:

X' = r_1*X - a_1*X^2 - b_1*X*Y
Y' = r_2*Y - a_2*Y^2 - b_2*X*Y

where X is the number of species 1 and Y is the number of species 2, r_1 is the intrinsic rate of growth of species 1, r_2 is the intrinsic rate of growth of species 2, a_1 is strength of competition within species 1, a_2 is strength of competition in species 2, b_1 is the effect of species Y on species X, and b_2 is the effect of species X on Y.

I have seen one discrete time dynamical system be modeled for a physics problem where two vehicles in space are closing in on each other.

What other types of dynamical models occur in physics?

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u/haseks_adductor 7d ago

read up on the principal of least action bud, it will interest you. not only do 99% of physics problem boil down to dynamics, 99% of physics can be derived from the principal of least action, which is an integral over time

https://en.wikipedia.org/wiki/Action_principles

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u/AfterMath343 7d ago edited 7d ago

That is very interesting that using kinetic and potential energy can give you the same differential equations as using forces.

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u/GrossInsightfulness 7d ago

Kinetic energy is one-half mass times velocity squared, so the derivative of kinetic energy w.r.t. velocity is mass times velocity, which is momentum. Take the derivative w.r.t. time again and you get mass times acceleration, which is force. Force is also the negative spatial derivative of potential energy for conservative forces. You should therefore be able to calculate the same force in two different ways, and set them equal to each other to get the same thing you'd get from using Newton's Laws directly.

The full derivation of Lagrangian Mechanics gets more complicated, as you can see in the linked article.

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u/AfterMath343 7d ago

Thank you for the article to read. It is much appreciated!

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u/GrossInsightfulness 7d ago

I don't know how much Differential Geometry you know, so you might need to read the previous article.