Motion in a Two-Dimensional Harmonic Potential

where , and . It follows that the particle is subject to a force,

(155) |

(156) |

where .

Since Equations (157) and (158) are both *simple harmonic equations*,
we can immediately write their general solutions:

(159) | |||

(160) |

Here, , , , and are arbitrary constants of integration. We can simplify the above equations slightly by shifting the origin of time (which is, after all, arbitrary):

(161) |

where . Note that the motion is clearly

Using standard trigonometry, we can write Equation (163)
in the form

(164) |

(165) |

Unfortunately, the above equation is not immediately recognizable as being the equation of any particular geometric curve:

Perhaps our problem is that we are using the wrong coordinates.
Suppose that we rotate our coordinate axes about the -axis by an
angle , as illustrated in Figure A.100. According to Equations (A.1277) and (A.1278), our old coordinates (, ) are related to our new coordinates
(, ) via

Let us see whether Equation (166) takes a simpler form when expressed in terms of our new coordinates. Equations (166)-(168) yield

We can simplify the above equation by setting the term involving to zero. Hence,

(170) |

(171) |

where

(173) | |||

(174) |

Of course, we immediately recognize Equation (172) as the equation of an

We conclude that, in general, a particle of mass moving in the two-dimensional harmonic potential (154) executes a *closed elliptical
orbit* (which is not necessarily aligned along the - and -axes), centered on the origin, with
period
, where
.

Figure 10 shows some example trajectories calculated for , , and the following values of the phase difference, : (a) ; (b) ; (c) ; (d) . Note that when the trajectory degenerates into a straight-line (which can be thought of as an ellipse whose minor radius is zero).

Perhaps, the main lesson to be learned from the above study of two-dimensional motion in a harmonic potential is that comparatively simple patterns of motion can be made to look complicated when expressed in terms of ill-chosen coordinates.