Given a system of periodic Ordinary Differential Equations of the form
![\begin{displaymath}
{d\over dt} \left[{\matrix{x\cr y\cr v_x\cr v_y\cr}}\right] ...
... 0\cr}}\right]\left[{\matrix{x\cr y\cr v_x\cr v_y\cr}}\right],
\end{displaymath}](f_1477.gif) |
(1) |
the solution can be written as a linear combination of functions of the form
![\begin{displaymath}
\left[{\matrix{x(t)\cr y(t)\cr v_x(t)\cr v_y(t)\cr}}\right] ...
...x{x_0\cr y_0\cr v_{x0}\cr v_{y0}\cr}}\right]e^{\mu t}P_\mu(t),
\end{displaymath}](f_1478.gif) |
(2) |
where
is a function periodic with the same period
as the equations themselves. Given an Ordinary
Differential Equation of the form
![\begin{displaymath}
\ddot x+g(t)x=0,
\end{displaymath}](f_1480.gif) |
(3) |
where
is periodic with period
, the ODE has a pair of independent solutions given by the
Real and Imaginary Parts of
Plugging these into (3) gives
![\begin{displaymath}
\ddot w+2i\dot w\dot\psi+w(g+i\ddot\psi-\dot\psi^2)=0,
\end{displaymath}](f_1489.gif) |
(7) |
so the Real and Imaginary Parts are
![\begin{displaymath}
\ddot w+w(g-\dot\psi^2)=0
\end{displaymath}](f_1490.gif) |
(8) |
![\begin{displaymath}
2\dot w\dot\psi+w\ddot\psi=0.
\end{displaymath}](f_1491.gif) |
(9) |
From (9),
Integrating gives
![\begin{displaymath}
\dot\psi={C\over w^2},
\end{displaymath}](f_1495.gif) |
(11) |
where
is a constant which must equal 1, so
is given by
![\begin{displaymath}
\psi=\int_{t_0}^t {dt\over w^2}.
\end{displaymath}](f_1497.gif) |
(12) |
The Real solution is then
![\begin{displaymath}
x(t)=w(t)\cos[\psi(t)],
\end{displaymath}](f_1498.gif) |
(13) |
so
and
which is an integral of motion. Therefore, although
is not explicitly known, an integral
always exists.
Plugging (10) into (8) gives
![\begin{displaymath}
\ddot w+g(t)w-{1\over w^3}=0,
\end{displaymath}](f_1504.gif) |
(16) |
which, however, is not any easier to solve than (3).
References
Abramowitz, M. and Stegun, C. A. (Eds.).
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, p. 727, 1972.
Binney, J. and Tremaine, S. Galactic Dynamics. Princeton, NJ: Princeton University Press, p. 175, 1987.
Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion. New York: Springer-Verlag, p. 32, 1983.
Margenau, H. and Murphy, G. M. The Mathematics of Physics and Chemistry, 2 vols.
Princeton, NJ: Van Nostrand, 1956-64.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 556-557, 1953.
© 1996-9 Eric W. Weisstein
1999-05-26