If I recall it correctly, algebraic curves (ie. curves defined by solutions of a polynomial) cannot be used to describe mechanical curves like the catenary (hanging chain), cycloid or other common curves like the Archimedian spiral. All of these counterexamples involve some sort of trigonometric or exponential functions, so I came to wonder
Is there an extension to algebraic geometry, which deals with these curves on a purely algebraic level?
I am thinking of something along the lines of an abstract notion of trigonometric function, formal power series or formal derivatives and derivations.
Here is a general framework. An algebraic ODE is an ordinary differential equation of the form $$ P(y, y', y'',...,y^{(n)})=b(t), $$ where $b(t)$ is the given (polynomial) function, $y=y(t)$ is the unknown function of one (real or complex) variable, $y^{(k)}$ is the $k$-th order derivative of $y$ and $P(x_0,...,x_n)$ is a polynomial of $n+1$ variables.
In what follows, I will be rather sloppy with domains of my functions since spelling this in all the detail is too painful
A special case of this is a linear algebraic ODE, which has the form $$ P_0(t)y(t) + P_1(t)y'(t)+...+ P_n(t) y^{(n)}(t)=b(t), $$ where each $P_i(t)$ is a given polynomial function.
One can generalize this notion to the setting of functions of several variables (algebraic PDEs): $y=y(x_1,...,x_n)$ is the unknown function, $P$ is a polynomial of many variables, $b=b(x_1,...,x_n)$ is the given polynomial function, the equation has the form $$ P(y,...,\frac{\partial^k y}{\partial x_{i_1} ... \partial x_{i_k}},....)=b(x_1,...,x_n). $$
One famous example: $y=y(x,t)$, $$ \frac{\partial y}{\partial t} + \frac{\partial^3 y}{\partial x^3} - 6\, y\, \frac{\partial y}{\partial x} =0. $$ Most (if not all) analytic functions one encounters in applied math, physics, math biology, etc., are solutions of algebraic ODEs or PDEs. In fact, it takes some effort to find any analytic function which is not a solution of an algebraic ODE (they do exist though). Whatever you construct using a "mechanical curve," I am sure will be a solution of an algebraic ODEs or PDEs.