What physic case can be answered in 5 ways?

Ayu ZM: What is a physic case that can be answered by 5 ways?
Please give me a physic case that van be answered by linear equation system, ordinary differential equation, interpolation, series, and eigen.
And the explanation of those five if you can.
Thank you before.

Answers and Views:

Answer by PhysicsDude
I think there are probably many physics problems that can be tackled using all the methods that you mentioned. The difference is that some methods will provide more accurate solutions, while others are approximations, and can only provide an order of magnitude answers.

For example, if we were to look at the hydrogen atom, which consist of one proton and one electron with the electron “orbiting” the proton. In this example, we can model it like a 2-body motion problem and use the standard circular motion equations such as:

θ = ωt
v = r·ω
a = r·ω²
F = mv²/r

The above are linear equations

Now we can model this system using differential equations that looks like:

ad²x/dt² + bdx/dt + c =0

For interpolation, we can model by getting a set of data points that represent parts of the solution to the actual equations of motion for the system.

For series, you approximate the true solution by using a series that if you take the limit, it approaches the true solution, but of course we can’t perform the operations of the series inifinite number of time, so we calculate the answer by only computing the first 2 components of the series.

For eigenvalues, we model the hydrogen atom as a quantum system that can only have certain states that are called eigenstates with eigenvalues that are solutions to the equation of motion (or the Hamiltonian).