mathematics//differential equations//ODE//oscillator

The typical second-order equation, and why it lives in a two-dimensional state space anyway.

The typical second-order equation, and why it lives in a two-dimensional state space anyway.

r˙=Aj,j˙=−Br⟹r¨+ABr=0.\dot r=Aj,\qquad \dot j=-Br\quad\Longrightarrow\quad \ddot r+ABr=0.r˙=Aj,j˙​=−Br⟹r¨+ABr=0.

Differentiate the first equation and substitute the second and the pair of first-order equations becomes one second-order equation. Read backwards, that is how a second-order equation is always turned into a first-order system: name the velocity as a state.

An acceleration is involved, but the acceleration is a function of the position, so it is not a new dimension. The state is (r,j)(r,j)(r,j), the plane is two-dimensional, and the trajectories are closed curves around the origin turning at angular frequency AB\sqrt{AB}AB​. Drawn in one quadrant, an arrow at (r,j)(r,j)(r,j) points with components (Aj,−Br)(Aj,-Br)(Aj,−Br): the arrows are the dotted variables, the axes are the plain ones (phase portrait).

The same equation with damping, r¨+cr˙+ABr=0\ddot r+c\dot r+ABr=0r¨+cr˙+ABr=0, spirals inward instead of circling; in the language of modes the undamped case has a purely imaginary pair of eigenvalues and the damped one a pair with negative real part.

Two initial data are needed, r(0)r(0)r(0) and j(0)j(0)j(0), one per state (initial condition); the energy-like quantity Br2+Aj2Br^2+Aj^2Br2+Aj2 is conserved by the undamped field, a constraint that appears from the equations rather than being imposed.