The subject is the 2-point boundary value problem for a second order ordinary
differential equation
formula here
the simplest and oldest of all boundary value problems, which arose in Euler's work
on calculus of variations in the eighteenth century. Accordingly, there is a vast
literature, mostly in the context of applied mathematics, where one frequently uses
the methods of functional analysis.
Here we adopt a different point of view: we look for the number of solutions of
problem (1), if any, and how this number varies with the endpoints
(t1, x1) and (t2, x2).
The concept of focal decomposition, introduced in [15] and developed by Peixoto and
Thom in [19], expresses precisely this point of view (see §2). It was further developed
by Kupka and Peixoto in [10] in the context of geodesics. From there, one is led
naturally to relationships with the arithmetic of positive definite quadratic forms, a
line that is considered in [16]. In both [10]
and [16] attention is drawn to the close
formal relationship between focal decomposition and the Brillouin zones of solid
state physics. In [17] this line is pursued further and it is pointed out that the focal
decomposition associated to (1) appears naturally as a prerequisite for the
semiclassical quantization of this equation via the Feynman path integral method.
The main goal of the present paper is to prove Theorem 2, stated at the end of §4.
There we give a functional
∫t2t1L
(t, x, x˙) dt and consider its Euler equation. To this
second order differential equation we associate the corresponding focal decomposition.
Theorem 2 gives a criterion on the Lagrangian L from which we get a rough
geometric description of this focal decomposition. It turns out to be somewhat similar
to the one associated to the pendulum equation x¨+sin x = 0,
worked out by Peixoto and Thom in [19, pp. 631, 197]. In the case
L = (x˙2/2)−V(x) this type of focal
decomposition is generic in some topological sense.