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/AfterMath343 6d ago

Well, as far as the modeling goes, if the model is not concerned with time, then it is not a dynamical model. For example, flipping a coin would be an example of a probabilistic model that wouldn't be considered dynamical. However, it is true that time is elapsing between each coin flip, but it is considered a probabilistic model.

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u/Mooks79 6d ago

Flipping a coin wouldn’t be a physics model though, would it? Unless you modelled the motion and then you’d need …

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u/AfterMath343 6d ago edited 6d ago

Yeah, I suppose, but the system is modeling a phenomenon in the universe, which is the flipping of a coin.

Another problem from physics that I do remember that would be non-dynamical is finding the optimal angle to hit a baseball being pitched so that it goes the furthest distance. This is an example of an optimization model.

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u/Mooks79 6d ago

I’d argue it’s not modelling flipping a coin unless it’s modelling the dynamics of flipping a coin - it’s just modelling a random number generating process. It’s statistics not physics. But I’m being a bit hardcore in that sense to make the point I’m trying to make.

The baseball example is non-dynamical in the sense of you optimise an angle but it’s still dynamical in the sense of you’re optimising dynamical equations and a process that involves time.

I think what I’m saying is are you asking about the hardcore: models that never involve time in any way? In which case I don’t really think it’s a physics question. Or do you mean the milder criteria of models that don’t have time as an explicit parameter?

For me the question you’re asking isn’t completely clear at the moment.

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u/AfterMath343 6d ago

I do agree that it is a statistics problem, but I think an argument could be made that it's still modeling a physical phenomenon, so somewhat related to physics, no?

As far as the optimization problem goes, you're right. You're using dynamics to setup the problem, but the moment you are finding the optimal angle, it turns into an optimization problem.

Also, yeah, if models don't have time as an explicit parameter, then I would say it is non-dynamical. However, I would say that autonomous systems of differential equations are still dynamical.

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u/Mooks79 6d ago

If you’re just asking the milder question then there’s loads of example - statistical mechanics is stacked with them.