Because , we can write

(7.11) |

Multiplying Equation (7.12) by , and then differentiating with respect to time, we obtain

Now, Furthermore, and where use has been made of Equation (7.14). Thus, it follows from Equations (7.13), (7.15), and (7.16) thatLet us take Equation (7.17), multiply by , and then sum over all . We obtain

(7.18) |

(7.19) |

It is helpful to introduce a function , called the *Lagrangian*, that
is defined as the difference between the kinetic and potential energies of the dynamical system under investigation:

According to the preceding analysis, if we can express the kinetic and potential energies of our dynamical system solely in terms of our generalized coordinates and their time derivatives then we can immediately write down the equations of motion of the system, expressed in terms of the generalized coordinates, using Lagrange's equation, Equation (7.22). Unfortunately, this scheme only works for conservative systems.

As an example, consider a particle of mass moving in two dimensions in the central potential . This is clearly a two-degree-of-freedom dynamical system. As described in Section 4.4, the particle's instantaneous position is most conveniently specified in terms of the plane polar coordinates and . These are our two generalized coordinates. According to Equation (4.13), the square of the particle's velocity can be written

(7.23) |

(7.25) | ||||

(7.26) |

(7.27) | ||

(7.28) |

(7.29) | ||

(7.30) |