We consider the ordinary differential equation
where , and are positive . It’s evident that the equations
which leads to the general integral
The equation (5) with the plus sign represents a family of smooth arcs once the rectangle . Every single arc two opposite sides of which are tangent lines of the arc. The situation is analogical in the case of the minus sign. One arc of both kinds passes through every interior point of .
For better examining the curves (5) one may take
for a parametre. Then, , and (5) may be replaced by
Letting to change freely, for any given value of the equations (6) represent a continuous curve formed by an arc from the first family and another arc from the second family. All such curves (6) are integral curves of the equation (2) and are called Lissajous curves.
It may be shown that for any rational value of , (6) is a smooth closed curve, except when the curve comes to a vertex of the rectangle . If the value of is irrational, then (6) is never a closed curve, and any such curve fills the whole rectangle in the sense that it comes arbitrarily to every point of . In the former case, all integral curves of (2) are algebraic (http://planetmath.org/AlgebraicFunction).
Some figures in http://de.wikipedia.org/wiki/Lissajous-FigurWiki
- 1 E. Lindelöf: Differentiali- ja integralilasku III 1. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
|Date of creation||2013-03-22 19:13:33|
|Last modified on||2013-03-22 19:13:33|
|Last modified by||rrogers (21140)|