My boyfriend is a huge nerd and likes to talk about physics. Most of the things he tells me I can understand but I still can’t grasp differential equations. I love talking about physics and programming with him and I just wanna understand. Thanks!
In: Physics
Surely you know about recurrence relation right?
Here is a simple example:
I am walking. Each second, I walk 2 steps. After 60 seconds, I will walk 120 steps.
“Each second, I walk 2 steps” is a simple example of a recurrence relation. It tells you the relationship between the number of steps between one point in time that 1 second afterward. Once you know the recurrence relation, you can attempt to make long-term prediction, such as “After 60 seconds, I will walk 120 steps”.
Recurrence relation is important, because a lot of time, short-term relationship is easy to describe, but long-term relationship are not. “Every year I gain 10% in compound interest” is something you might heard, but you have to put in effort to figure out how much money you will gain after 10 years.
Now, differential equation is very similar to recurrence relation. The difference between them is that recurrence relation can describe how things change between discrete step in time (like “after 1 day”), but differential equation is needed to describe how things change continuously over time. This introduces some additional difficulty, because there are no “next” step, so differential equation only describe how things change over an arbitrarily short interval. For the purpose of simulation, people often treat differential equation as recurrence relation anyway (by picking a very short time interval and pretend that the equation is talking about discrete time step).
Why is differential equation important in physics? Because time is believed to be continuous, and physical law are described in term of relationship between short-term changes in physical quantity. For example, Newton’s law of gravitation describe how short-term change of objects’ positions are related to objects’ mass and distance. And the reason why we focus on short-term relationship is because these are the one that we can experiment on and understand intuitively.
Once you have a differential equation, you can attempt to solve them. It’s usually very hard, however.
There *are* physical laws that does not rely on short-term changes, but instead focus on long-term relationship between physical quantity. But generally they are derived *after* we have our differential equation, not before. Also, they are less intuitive to understand. For example, Lagrange’s mechanics posit that an object will choose a path that minimize its action. This is not intuitive to us, because it seems to say that an object has a will of its own and can see the future. But Newton mechanics is intuitive: things push on each other, inflict force, and force change velocity. But this require differential equation.
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