# Lagrangian Mechanics Made Simple

Part 1

Introduction

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

Lagrangian mechanics is mathematically equivalent to the usual Newtonian approach of "apply forces to things and see how they move." Instead of examining the forces on a body directly, it looks at the kinetic and potential energies of a system of objects. This approach simplifies many complicated problems, including things like waves in continuous media and orbital mechanics.

Since this series is meant to be of interest to game designers, we will initially focus on orbital mechanics. Later installments in this series will discuss motion in a rotating coordinate system, the Coriolis force, Kepler's equation for planetary orbits, and the stability of objects at the fourth and fifth Lagrange points. If there is sufficient interest, the series may be expanded to include other topics.

# Basics and Notation

The reader is assumed to have some grasp of calculus, and at least a passing knowledge of what is meant by such quantities as velocity, acceleration, momentum, and kinetic and potential energy.

We adopt the usual physics convention of using a "dot" to represent
a time derivative. That is, instead of writing

(1) |

(2) |

(3) |

Since we will be dealing with multiple variables, we also use
partial derivatives,

(4) |

# Lagrange's Equation

The core of Lagrangian mechanics is a quantity called the
Lagrangian. Consider a system with one possible direction of
motion, denoted by the position variable . The kinetic energy
may, depending on the coordinate system used, depend on both the
position and velocity of the object, so we write it as .
Similarly, we write the potential energy of the system as .
The Lagrangian is given by the equation

(5) |

# Some Simple Examples

To get some idea of what this means, consider the case of an object
with height falling under the influence of gravity. The kinetic
energy of the object only depends on the velocity ,

(7) |

(8) |

(9) |

(10) |

which shows that the object is falling with acceleration . The minus sign indicates that the object is falling down, instead of up.

The above example can be generalized to motion in an arbitrary potential
. In this more general case, the equation of motion (11)
is replaced by the more general

(12) |

^{1}

These examples are fairly basic, and don't really reveal the full power of the Lagrangian. We'll examine some more interesting cases in the next installment.

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