Tatiana Cook

2022-10-07

differential equations and physical intuition

Often when you study differential equations, you find phenomena in nature modeled by those equations. Sometimes an insight into a physical problem can help you to solve a differential equation. My question is: If you are a pure mathematician studying differential equations, do you have to be good at physics (biology, finance) too?

Often when you study differential equations, you find phenomena in nature modeled by those equations. Sometimes an insight into a physical problem can help you to solve a differential equation. My question is: If you are a pure mathematician studying differential equations, do you have to be good at physics (biology, finance) too?

Rihanna Blanchard

Beginner2022-10-08Added 13 answers

Step 1

By looking at physical phenomena, you'll see an application of a (partial) differential equation expressed in a particular coordinate system, or with particular boundary conditions.

The solutions for these applications will likely be a subset of a more general solution.

Take Gauss's Law:

$$\overrightarrow{\mathrm{\nabla}}\cdot \overrightarrow{E}(\overrightarrow{r})=4\pi \rho (\overrightarrow{r}).$$

Solving this differential equation for the electric field $\overrightarrow{E}$ might be made much easier if you consider that the electric field is derived from a scalar potential function that depends only on the magnitude of $\overrightarrow{r}$ . Then you can choose your surface of integration to make the dot product trivial (a sphere centered at $\overrightarrow{r}=0$). Then your partial differential equation becomes a relatively easy regular differential equation in one variable.

Step 2

But this might be a bit divorced from what you need or want. Physicists use differential equations to explain something: electromagnetic fields, mechanics, heat flow, etc. Once they have the model (the equation) they apply it, and see if it matches reality. If you're staying within the realm of mathematics, then you might not want to do that, but instead rely on formal proof, or other "things you know" from related areas of mathematics. Using physical intuition in pure mathematics might lead you to a wrong conclusion, because your application might not be correct.

By looking at physical phenomena, you'll see an application of a (partial) differential equation expressed in a particular coordinate system, or with particular boundary conditions.

The solutions for these applications will likely be a subset of a more general solution.

Take Gauss's Law:

$$\overrightarrow{\mathrm{\nabla}}\cdot \overrightarrow{E}(\overrightarrow{r})=4\pi \rho (\overrightarrow{r}).$$

Solving this differential equation for the electric field $\overrightarrow{E}$ might be made much easier if you consider that the electric field is derived from a scalar potential function that depends only on the magnitude of $\overrightarrow{r}$ . Then you can choose your surface of integration to make the dot product trivial (a sphere centered at $\overrightarrow{r}=0$). Then your partial differential equation becomes a relatively easy regular differential equation in one variable.

Step 2

But this might be a bit divorced from what you need or want. Physicists use differential equations to explain something: electromagnetic fields, mechanics, heat flow, etc. Once they have the model (the equation) they apply it, and see if it matches reality. If you're staying within the realm of mathematics, then you might not want to do that, but instead rely on formal proof, or other "things you know" from related areas of mathematics. Using physical intuition in pure mathematics might lead you to a wrong conclusion, because your application might not be correct.

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