Towards the end of the previous article, we had arrived at an important result called the D’Alembert principle. The principle describes the equilibrium condition for a mechanical system, without the need for the inclusion of the constraint forces. For every force that threatens to change the momentum of a particle, there exists an opposing force that can keep the body in equilibrium. Therefore, any situation where mechanical equilibrium exists, we can view the set up as a balance between an applied force and an opposing -force. This reasoning helps us figure out what are the conditions that keep a system in mechanical equilibrium. Our job now is to incorporate the generalized coordinates and obtain new concepts such as the generalized force, which ultimately help us describe the motion of objects.
Force to generalized force
Now, if the coordinates being used here are responsible for this equation to appear fussy or complicated (in a given problem), we know that we can always switch to other coordinates with the help of the transformation equations between the coordinates. For e.g., if we chose to switch to some generalized coordinates , we have at our rescue the transformation equations,
and so on.
Or, in short hand notation, we shall write them as,
Then the virtual displacement, which just is an infinitesimal variation of the coordinate at a frozen instant of time , can be written as (using the chain rule),
Replacing with this expansion, in equation (1), we can write the virtual work as,
as the components of a “generalized force”.
(Observe that just like generalized coordinates needn’t be measured in meters, the generalized force needn’t be necessarily measured in newtons. But the product of the generalized force and the generalized coordinates must be in joules or any equivalent unit of work.)
With this, we can comfortably define virtual work as a product of the generalized forces and the generalized coordinates.
We’ve successfully expressed the first term of equation (1) in terms of the generalized coordinates. We have to do the same with the second term .
The other term
Now, say we want to calculate the time derivative:
The second term in the parenthesis on the right side here contains a derivative of first with respect to and then with respect to . We could as well interchange the order of these differentiations and write,
Where we have used the definition of the Cartesian velocity components: ; since the ‘s are a function of the ‘s. This equation here also implies that . Now, putting all these results into equation (5), we have,
Introducing the kinetic energy
Closely observe the first term in the parenthesis here, does it not look like the expansion,
So, we shall rewrite equation (7) as,
Here, is just the famous classical kinetic energy , of the masses. Hence, we may write,
Another way of writing this is,
The virtual displacement was chosen arbitrarily (nothing particular about it was assumed), and so for the above equation to hold for a non-zero displacement, the term in the parenthesis must vanish:
We have here a recipe for obtaining the equation of motion of any object moving due to a force. Knowing the forces and using the relation (2), , we can know the generalized forces. Then all that remains is for us to calculate the kinetic energy of the object and put it all in the above equation (8), and we know how the object moves.
In the following article, we will see how the concept of the generalized force we understood here, can be used to deduce a set of marvelous equations called the Lagrange equations, which form an alternate explanation of object motions.