De Alembert’s Principle and Generalized Force, a technical discourse on Classical Mechanics

A technical discourse on formal classical mechanics. This is a 12 slide introduction to the basics of how Newton's Laws are generalized into a Lagrangian Dynamics apt at the level of an advance student of Physics.

1. D’ Alembert’s Principle and Generalized Force. “A lecture on basics of classical mechanics .” M. S. Ramaiah University of Applied Sciences . Faculty of Science and Humanities Manmohan Dash

2. Virtual Displacement We define a Virtual Displacement of a system of particles described by the coordinates r i at a given instant of time t. Lets say the system has an infinitesimal amount of change in its configuration due to an arbitrary change in its coordinates;  r i , in accordance with the applied forces F i (a) and constraint forces f i .

3. Virtual Displacement A virtual displacement is different from an actual displacement in that in actual displacement there is a time lapse dt during which applied forces and constraint forces may change. If the system is in equilibrium the force on the i th particle is zero, so F i = 0. As a result virtual work done on the i th particle is zero or F i .  r i =0. Considering all particles;  i F i .  r i = 0

4. Principle of virtual work The total force on i th particle can be written into the applied force and the constraint force . So; F i = F i (a) + f i From  i F i .  r i = 0;  i F i (a) .  r i +  i f i .  r i = 0 . W e assume the “ net virtual work done by the forces of constraints ” to be zero so that  i F i (a) .  r i = 0 . This is known as “ Principle of virtual work ”.

5. Virtual displacement In the above equation the virtual coordinates  r i are not independent of each other and as a consequence all of the applied forces are not zero. The virtual coordinates  r i need to be transformed into generalized coordinates q i which are then independent of each other.

6. Conditions on Virtual Work Under what conditions  i f i .  r i = 0 is valid; Rigid Body constraints are one such example. A particle moving on a surface. Here the forces of constraints and displacement of particle are perpendicular to each other. Rolling friction, as a constraint force; during virtual displacement the point in contact with surface is momentarily at rest .

7. Conditions on Virtual Displacement Under what conditions  i f i .  r i = 0 is valid; If the surface on which a particle is in motion is moving , the condition that “zero virtual work due to forces of constraints” is still valid in the instant of motion.

8. Equation of Motion The condition of virtual work being zero is not sufficient for a general description of motion We need to introduce the equation of motion ; F i = t p i where the t stands for differentiation wrt time. We read t p i as p i dot . We thus have F i - t p i = 0 . It states that the system of particles is in a state of equilibrium under the application of a force F i and a reverse effective force - t p i .

9. D’ Alembert’s Principle. The principle of virtual work now becomes;  i ( F i - t p i ).  r i = 0. Since F i = F i (a) + f i this becomes;  i ( F i (a) - t p i ).  r i +  i f i .  r i = 0. By applying “zero net virtual work by the forces of constraints” as discussed already we have;  i ( F i (a) - t p i ).  r i = 0.  i ( F i (a) - t p i ).  r i = 0 is known as “ D’ Alembert’s Principle “.

10. Generalized Coordinates. We transform the ordinary “ dependent coordinates r i ” into holonomic “ generalized coordinates q j ” which are independent of each other; r i = r i (q 1 , q 2 , q 3 , q j , …, q n , t) T here are n such generalized coordinates. (and N dependent coordinates r i ).

11. Generalized velocity and displacement. We apply the chain rule of partial differentiation to get the generalized velocity . v i  d r i / dt =  k ∂ r i /∂ q k . t q k + ∂ r i /∂t. This shows ordinary velocity v i is related this way to the generalized velocity t q k . Also arbitrary virtual displacement  r i is connected to the generalized virtual displacement  q j ;  r i =  j ( ∂ r i /∂ q j ) . q j

12. Generalized work and force. Virtual Work of F i ; we dropped index of F i (a)  i F i .  r i =  i , j F i . ∂ r i /∂ q j  q j =  i Q j  q j with Q j =  i F i . ∂ r i /∂ q j The q is generalized coordinates, Q is “ generalized force ” and  q are “generalized virtual displacement”. Also Q  q are “ generalized work ”. Q’s and q’s do not have the dimension of force and length respectively but Q  q have the dimension of work necessarily.