The spring equation and the shape of simple motion

The spring is the equation everything else in physics gets compared to, so it’s a good place to start looking at phase space. It comes from about the least surprising fact you can state about a spring: the harder you pull it, the harder it pulls back. Stretch it twice as far and it fights you twice as hard. Write that down and you get Hooke’s law, $F = -kx$, where $x$ is how far the mass has been pushed from its resting spot and $k$ measures how stiff the spring is. The minus sign is the whole story. Whichever way you shove the mass, the force points back the other way.

Newton says force is mass times acceleration, and acceleration is the second derivative of position, so the sentence “the force always points back toward the middle” turns into

$m\ddot{x} = -kx.$

That’s the spring equation. To draw it I split that one second-order equation into two first-order ones by giving the velocity its own name. Let $v = \dot{x}$. Then the state of the spring at any instant is the pair $(x, v)$, and the two of them chase each other around:

$\dot{x} = v, \qquad \dot{v} = -\tfrac{k}{m}\,x.$

Position drifts in the direction the velocity points, and velocity gets pulled back toward zero in proportion to position. That pair is exactly the current the widget draws.

interactive · the spring equation in phase spaceopen ↗

Leave the damping slider at zero and the flow settles into a set of nested loops going around the origin, all of them closed. Those loops are ellipses, and the reason they close is that a frictionless spring never loses anything. The quantity

$E = \tfrac{1}{2}v^2 + \tfrac{1}{2}\tfrac{k}{m}x^2$

is the energy, kinetic plus the energy stored in the stretch, and it doesn’t change as the mass swings. Set it to a constant and you’ve written the equation of an ellipse, which is why I drew a few of them faintly on top of the flow. Each streak is trapped on the loop it started on, forever, tracing the same swing. Turn up the stiffness and the ellipses get taller because a stiffer spring whips the mass through the middle faster, so the same swing carries more speed. The middle point itself never moves. Sit the mass exactly at rest at the center and it stays there, which is why that dot is a fixed point, and because everything nearby just circles it and never falls in, it’s the kind physicists call a center.

Real springs lose energy, so slide the damping up and the picture changes character completely. Now the mass drags against something and the equation grows a term that always opposes the velocity:

$\ddot{x} = -\tfrac{k}{m}x - \tfrac{c}{m}\dot{x}.$

The closed loops break open. Every streak still circles the origin but each time around it comes back a little closer, and the whole plane turns into a spiral draining into the center. That’s a ringing note fading out, drawn as a picture. Push the damping high enough and the spiral stops being a spiral at all. The mass no longer overshoots, it just leans back to the middle and stops, and the flow slides straight in without a single loop. The crossover between those two behaviors is critical damping, the setting a screen door closer is tuned to so the door swings shut fast without slamming or bouncing.

None of this needs the solution formula. You never have to write down a sine or an exponential to see that a frictionless spring rings forever and a damped one rings down. It’s sitting right there in whether the loops close or spiral, which to me is the nicest argument for looking at these things as flows in the first place.