What are governing equations?
Terminology
In systems given such as a dynamical system, an equation made up of variables that describe the state of the system is called a governing equation.
Explanation
$$ x_{t+1} = r x_{t} (1 - x_{t}) $$ For example, suppose the state “rabbit population” is represented by the variable $x_{t}$. When such a system can be described by the logistic map as above, the above map describing the dynamical system may be called the governing equation.
Because governing equations often deal with complex systems, they are frequently expressed as differential equations. For instance, the SIR model used to model infectious diseases adopts the following ordinary differential equations as its governing equations. $$ \begin{align*} {{d S} \over {d t}} =& - {{ \beta } \over { N }} I S \\ {{d I} \over {d t}} =& {{ \beta } \over { N }} S I - \mu I \\ {{d R} \over {d t}} =& \mu I \end{align*} $$
More complexly, in areas such as fluid dynamics, the Euler equations serve as governing equations in the form of the following partial differential equations that describe the flow velocity using the variable $\mathbf{u}$. $$ {\frac{ D \mathbf{u} }{ D t }} = - {\frac{ 1 }{ \rho }} \nabla P + \mathbf{g} $$