More examples of equations that unexpectedly model nature in similar ways

[syfry]

Been dipping my toes into maths by examining how equations work on the most basic level, and I love encountering equations that turn out to model similar aspects in nature, for example the inverse square law is apparent in equations for gravity and for electromagnetism.

In the thumbnail of this video, the equations for electric and gravitational forces are very similar.

Symmetry (visual) might be the description I'm going for.

What are more examples of equations in science with visual (or functional) type of symmetry?

 

[Mark44]

I'm not totally clear on what you're asking, but will assume that by "symmetry" you mean equations for physical quantities that are analogous to each other. If so, here are a couple that come to mind.
Linear force F and acceleration -- F = ma
Rotational torque and rotational acceleration ##\tau = I\alpha##
In these equations the pairs force (F) and torque (##\tau##), mass (m) and moment of inertia (I), and acceleration (a) and rotational acceleration (##\alpha##) are analogous.

Another example that is more complicated is how an LRC (inductor, resistor, capacitor) electrical circuit can be described by essentially the same second-order differential equation as a damped, spring and mass system. The equation for the electrical circuit, where the source voltage is constant is ##\ddot I(t) + \frac R l \dot I(t) + \frac 1 {LC} I(t) = 0##. Here I(t) is the current at time t, R is the resistance of the resistor, L is the inductance of the coil, and C is the capacitance of the coil.

The analogous equation for a damped, spring mass system is ##\ddot x(t) + \frac c m \dot x(t) + \frac k m x(t) = 0##.

 

[hutchphd]

I am always aware of the adage "If you have a hammer, all your problems look like a nail" which I believe is also called "Maslow's Hammer".
There are only a small number of equations that we can happily solve and so our musings lead us there. Of course that this is true may be further indication of the fundamentals of the solution.....

 

[gmax137]

There are only a small number of equations that we can happily solve and so our musings lead us there.

One of my professors was musing in class one day. "We see this - second order partial differential equations - everywhere, in all kinds of unrelated places. Does this tell us more about the world, or more about ourselves?"

 

[syfry]

I'm not totally clear on what you're asking, but will assume that by "symmetry" you mean equations for physical quantities that are analogous to each other. If so, here are a couple that come to mind.
Linear force F and acceleration -- F = ma
Rotational torque and rotational acceleration ##\tau = I\alpha##
In these equations the pairs force (F) and torque (##\tau##), mass (m) and moment of inertia (I), and acceleration (a) and rotational acceleration (##\alpha##) are analogous.

Another example that is more complicated is how an LRC (inductor, resistor, capacitor) electrical circuit can be described by essentially the same second-order differential equation as a damped, spring and mass system. The equation for the electrical circuit, where the source voltage is constant is ##\ddot I(t) + \frac R l \dot I(t) + \frac 1 {LC} I(t) = 0##. Here I(t) is the current at time t, R is the resistance of the resistor, L is the inductance of the coil, and C is the capacitance of the coil.

The analogous equation for a damped, spring mass system is ##\ddot x(t) + \frac c m \dot x(t) + \frac k m x(t) = 0##.

Yeah that's the word, analagous!

Really nice example with torque. The examples with the electrical circuit also, so symmetrically satisfying and interesting.