Generalized Kinetic Energy

We start from the definition of kinetic energy. 

2. Write the displacement in Generalized coordinates. 

ri=ri (q1,q2,....,qn,t)

3. We take differential of the Generalized displacement and simplify the expression. The terms with displacement q separately and term with time separately. 

Image 1

4. Now, we divide the differential with dt in both sides. This gives ri dot.

5. We take the square of the differential in step 4.

Image 2

6. We apply scheleronomic constraint here. Which was defined in the blogpost. Which means t=0 ans term containing t is neglected.

7. Now put the expression of r dot square in the expression of kinetic energy and rearrange.

Image 3

8. Now suppose the term in paranthesis as ajk.

9. Multiply both sides of equation with term containing partial derivative of ql dot, as shown in last step of image.

Image 4

10. We open the partial derivative by product rule.

11. We define kronicker delta function and apply it on the equation.

12.

Kronicker delta is a tensor, when applied on vector, it contracts the same induces. 

Image 5

13. We multiply equation with summation ql dot. And get an expression.

14. We combine the terms. In one term we replace l by k, in second we replace l by j. This is to get while expression in similar indices (j,k in our case)

Image 6 

15. On simplifying, we get an expression of generalized kinetic energy as shown. 

Image 7,8