In functional analysis the variable itself is a function.
This is used e.g. in the Lagrangian formulation of mechanics where one derives the Lagrangian i.e.
L = kinetic energy – potential energy.
The variables, or degrees of freedom, can be selected to make the problem as easy as possible. They can be cartesian coordinates, velocities or momentums for example.
By solving Lagrange’s differential equation the Lagrangian can be found.
A similar system was devsed by William Rowan Hammilton. He studied the hamiltonian for the system. This is the sum of the kinetic and potential energy of the system.
It is used for example in the Schrödinger equation.