, , k =1,2,….. n............................................ (2.5.7)

one obtains the following equation.

............................................................ (2.5.8)

Now,

................................(2.5.9)

Substituting (2.5.9) in (2.5.8) we have

.................................................................... (2.5.10)

Considering the arbitrariness of the virtual displacement , equation ( 2.5.10) will be satisfied for all values of provided

........................................................................(2.5.11)

Equation (2.5.11) is known as Lagrange's equation .

Considering both conservative force and nonconservative force , the total generalized force can be written as

.............................................................................................................(2.5.12)

and recalling potential energy depends on coordinates alone, the work done by the conservative force is equal to the negative of the potential energy V. Hence, one may write

.................................................................... (2.5.13)

So the conservative generalized forces have the form

, k =1, 2,….. n ...........................................................................................(2.5.14)

Substituting Eq. (2.5.12) and Eq. (2.5.13) in Eq. (2.5.10) we have