for , where are each equal to the mass of the first particle, are each equal to the mass of the second particle, et cetera. Furthermore, the kinetic energy of the system can be written

Now, because , we can write

(B.11) |

for . Hence, it follows that . According to the previous equation,

where we are treating the and the as independent variables.

Multiplying Equation (B.12) by , and then differentiating with respect to time, we obtain

Now,

Furthermore,

and

where use has been made of Equation (B.14). Thus, it follows from Equations (B.13), (B.15), and (B.16) that

(B.17) |

Let us take the previous equation, multiply by , and then sum over all . We obtain

(B.18) |

where use has been made of Equations (B.9) and (B.10). Thus, it follows from Equation (B.6) that

(B.19) |

Finally, making use of Equation (B.8), we get

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

Because the potential energy, , is clearly independent of the , it follows from Equation (B.20) that

for . This equation is known as

According to the previous 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, (B.22). Unfortunately, this scheme only works for energy-conserving systems.