Variations of Hodge structure of weight two are integral manifolds for a distribution in the tangent bundle of a period domain. This distribution has dimension
h
2
,
0
h
1
,
1
{h^{2,0}}{h^{1,1}}
and is nonintegrable for
h
2
,
0
>
1
{h^{2,0}} > 1
. In this case it is known that the dimension of an integral manifold does not exceed
1
2
h
2
,
0
h
1
,
1
\frac {1} {2}{h^{2,0}}{h^{1,1}}
. Here we give a new proof, based on an analogy between Griffiths’ horizontal differential system of algebraic geometry and the contact system of classical mechanics. We show also that any two points in such a domain can be joined by a horizontal curve which is piecewise holomorphic.