Generalized Force

Photo by Abhishek Chandra on Unsplash

In the blog post, we discussed why the Generalized coordinates are necessary. Here we find, Generalized force’s expression.

Consider a particle that movers from r to an infinitesimal displacement.

There’s an infinitesimal work done.

Steps involved

1. We have dW = F. dr. We transform it regarding infinitesimal displacement. 

2. Write force and dr as vectors with a dot product in between them.

3. Upon multiplication, we get dW=Fxdx+Fydy+Fzdz and write it in terms of infinitesimal displacement.

4. We consider one dimensional case so only Fxdx is considered in the work done.

5. We write x in terms of Generalized coordinates.

6. The terms involving dummy index i are taken within brackets with summation i and the term with j are kept aside.

7. The term in parenthesis becomes the Generalized force expression. So we write the work done in terms of Generalized momentum.

Where i and j are dummy indices that are used as per choice.