# Ordinary differential equations as variational problems

Considering an ordinary differential equation of first order in the implicit form

$$F(q(t),\dot q (t))=\alpha,\,\,\, q(0)=q_0$$

with $\alpha,\, q_0$ constants, what is the relation of the solution $q(t)$ to solutions of the variational problem for $S[y]$, obtained by taking $F$ (or a suitable function thereof, e.g. $F^2$) as a Lagrange function

$$S[y] = \int_0^T F(y(t),\dot y (t))\, dt, \\ y(0)=q_0, \, \, \, y(T)=q(T)$$

where $T$ is a conveniently chosen positive constant.

$\bf{Corrected:}$

The solutions of the variational problem for $F(y(t),\dot y (t))$ imply another implicit equation

$$F(y(t),\dot y (t))-\dot y(t) \frac{\partial F}{\partial {\dot y(t)}} = \it{const.}$$

With $\dot y(t) = z$, the correct function $G(y,z)$ to take in the variational problem should therefore satify

$$\frac{\partial}{\partial z} \left( \frac{G}{z} \right) =\frac{F(y,z)}{z^2}$$

which relates $G$ to $F$ as a Lagrangian relates to its corresponding Hamilton function.

It seems therefore that a variational problem can be associated to any implicit, first order ODE.

• It is often the opposite: your equation may come from a variational problem, but it would be tortuous to vary what was already varied. Simpler answer: in general there is no relation, unless you take some integral of the equation. – John B Jan 29 '16 at 17:50
• @Jonas by integral of the equation you mean an integral of motion, like the Hamilton function (energy, conserved for a time-independent ODE). You are right, and I have now included this case. – Lupercus Jan 31 '16 at 18:45
• Yes, that's what I meant. For example $F$ may have been obtained from some variational problem, but you are talking about using it for kind of a "second" variational problem. – John B Feb 1 '16 at 17:27