What is the meaning of the term "prolongation" in differential geometry? Differential geometers often talk about "prolonging" a system of differential equations, or jet prolongation of bundle sections, but I don't really understand what mental picture the term "prolongation" is supposed to convey. Is it because when you introduce new variables for higher derivatives in a differential equation the system becomes "longer" when you write it down? Is that all there is to it, or is there some better reason for the terminology?


Actually not the system is becoming "longer" but the space of dependent variables. Basically, you want a geometric object not just representing the dependent and independent variables but also the appearing partial derivatives. So you "prolong" the space of dependent variables $U$ by spaces representing the partial derivatives of order $n$ denoted by $U_n$. Then $U^{(n)} = U \times U_1 \times \dots \times U_n$ is the $n$th prolonged space.

If you have a smooth function $u = f(x)$ with $f \colon X \to U$ then its $n$th prolongation is $u^{(n)} = \mathsf{pr}^{(n)} f$ given by the partial derivatives up to order $n$. For example the $2$-prolongation of a function $f(x,y)$ would be $(f, \frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial^2 f}{\partial x^2}, \frac{\partial^2 f}{\partial x \partial y}, \frac{\partial^2 f}{\partial x^2})$, representing the Taylor polynomial of second order. So the term prolongation originates from the prolongation of the space of dependent variables.

See also the book of Olver: "Applications of Lie Groups to Differential Equations", Sec. 2.3 and 3.5.

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