the mechanical-electrical analogy
Two utterly different physical setups — a weight bouncing on a spring and current sloshing in a wired loop — turn out to obey the very same equation. The analogy is the dictionary that translates between them, so that anything proved in one world is automatically true in the other. It is one of the most useful coincidences in applied mathematics.
Lay the two equations side by side. Mechanical: m x'' + c x' + k x = F(t). Electrical: L q'' + R q' + (1/C) q = E(t). Matching term by term gives the dictionary: mass m corresponds to inductance L, damping c to resistance R, spring stiffness k to inverse capacitance 1/C, displacement x to charge q, velocity x' to current q', and applied force F to applied voltage E. Whatever symbol you start with, you can rewrite the whole problem in the other language without changing a single number's role.
This is why engineers build electrical circuits to simulate mechanical systems (cheaper and safer than shaking a real bridge), and why the words natural frequency, damping ratio, resonance, and quality factor mean the same thing in both fields. The analogy is exact for the linear model: it holds precisely as long as both systems are well described by their constant-coefficient second-order equation.
A mechanical resonance at 5 Hz on a test rig can be reproduced by an RLC circuit whose L, R, and 1/C are chosen so L q'' + R q' + (1/C) q matches the spring-mass-damper numbers — the circuit peaks at the same 5 Hz.
Same equation, same behaviour: the electrical model is a stand-in for the mechanical one.
There is a second, equally valid dictionary (the force-current analogy) where mass pairs with capacitance instead of inductance. Neither is more correct; they suit different network topologies. State which convention you are using to avoid confusion.