= Mathematical model
{wiki}
A mathematical model represents selected features of a system with mathematical objects and relations so that its behavior can be analyzed or predicted. Choosing the representation is part of the work: specify the quantities tracked, the equations or <probability distributions> connecting them, and the conditions under which those connections are expected to apply. A <mathematical proof> can establish a consequence of the model; comparing that consequence with observations tests its suitability for the intended use.
For a <simple pendulum>, the angle $\theta(t)$ and <gravitational acceleration> $g$ give the ideal equation $\ddot\theta+(g/\ell)\sin\theta=0$. Its assumptions include a point mass, fixed <length> $\ell$, uniform <gravitational acceleration>, and no <damping>. The <small-angle approximation> gives the simpler <harmonic oscillator equation> and <oscillation period> $T_0=2\pi\sqrt{\ell/g}$. A larger <amplitude> changes the full model's period; drag requires a <damped pendulum>. The simpler model remains useful when its omitted effects are small for the question being asked.
A model need not track every detail. A <Bernoulli process> models repeated success/failure observations using one <probability> and an independence assumption; a <Gaussian process> specifies a <mean> and a <covariance function> to model related measurements. Each representation enables calculations by retaining some distinctions and omitting others. Useful simplification therefore comes with a stated domain of use and quantities against which its predictions can be checked.
Back to article page
