# Lagrangian Mechanics Made Simple

Part 3

Central Force Motion

**Ron Steinke <rsteinke@w-link.net>**

This month we examine the mechanics of a system of two particles which are attracted by a force which depends on the distance between them. This, when combined with our discussion of rotating coordinate frames last month, will allow us to begin our treatment of orbital mechanics in the next installment.

# Two Particles in a Central Potential

Consider two particles at positions
and ,
attracted by a force that depends on the distance

(1) |

(2) |

(3) |

We want to simplify the problem by removing the motion of the center of mass of
the system. In a system where there are no external forces acting on the
particles, the center of mass of the system,

(4) |

(5) |

(6) |

and the Lagrangian is

The Lagrangian is a sum of terms containing only and terms containing only . This means that and don't appear in each others' equations of motion. A Lagrangian with this property is said to be

**separable**, and can be written as a sum of two Lagrangians each governing part of the system. From (7), we see that the equation of motion for is

(8) |

# The Central Potential

The Lagrangian for the separation
between the particles is

is called the

**reduced mass**of the system. Solving the equation of motion for the particle separation is equivalent to solving the equation of motion for a particle of mass which is attracted to the point by an external force. Notice that if the mass of the first particle is much larger than that of the second (e.g., if the particles are the Earth and a satellite orbiting it), the denominator of (10) is approximately equal to , and the reduced mass is approximately equal to . This reflects the fact that, in the case of a large mass difference, the heavier particle is "fixed" and the lighter one orbits around it.

We can deduce several things about the particle motion without knowing the
exact form of .
The equation of motion derived from (9) is

(11) |

Since the vector
lies in a fixed plane, it can be described by its magnitude
and an angular displacement .
In terms of these variables, the Lagrangian is

(12) |

(13) |

(14) |

We can obtain a similar equation using energy conservation. The energy of the
system is

(15) |

(16) |

Lagrange's equations tell us that

(18) |

so the terms in the second set of brackets in (17) cancel. This leaves

(19) |

which is zero. This result is true in general when the kinetic energy depends quadratically on the velocities of the particles.

# The Equations Of Motion

The equations of motion for
have been reduced to conservation of energy,

(20) |

(21) |

(22) |

(23) |

In addition to finding
and
as functions of the time ,
it is also possible to find and equation relating the two,

(24) |

Current Issue: February 2003, Recent Issues: January 2003, November 2002, October 2002 |