for , where are each equal to the mass of the first particle, are each equal to the mass of the second particle,

Now, since
, we can write

(602) |

where we are treating the and the as

Multiplying Equation (603) by , and then differentiating
with respect to time, we obtain

Furthermore,

and

where use has been made of Equation (605). Thus, it follows from Equations (604), (606), and (607) that

(608) |

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

(609) |

(610) |

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:

for . This equation is known as

According to the above 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, (613). Unfortunately, this scheme only works for conservative systems. Let us now consider some examples.