Virtual Work - Is the presentation in Cornelius Lanczos wrong?

Question

Book: The Variational Principles of Mechanics by Cornelius Lanczos

Edition: 4th

Chapter: 3, The Principle of Virtual Work

I am on the second page of the 3rd chapter (pg 75; it has the Eqn. 31.1).

• Doubt 1: The author says $n$ external forces ($ {\vec{F_1}}, {\vec{F_2}} ......{\vec{F_n}} $) are applied on the system at $n$ points. Later he assumes that the no. of generalized coordinates is also $n$ (when he lists them as $(q_1, q_2 ....q_n)$. Why does this have to be true? Why does the number of external forces being applied to a system have to be equal to the no. of generalized coordinates?

• Doubt 2: On the same page the author further says:

"This is the reason why in the variational treatment of mechanics the "forces of constraint" which maintain certain given kinematical conditions are neglected, and only the work of the" impressed forces" needs to be taken into account."

The above sentence suggests that by 'impressed forces' the author means 'non-constraint forces'. That's alright. This is what standard textbooks say. But then he says:

"Now the principle of virtual work asserts that the given mechanical system will be in equilibrium if, and only if, the total virtual work of all the impressed forces vanishes: $$ {\bar {\delta w}}= {\vec {F_1}}.\delta {\vec {R_1}} + {\vec {F_2}}.\delta {\vec {R_2}}.......{\vec {F_n}}.\delta {\vec {R_n}}=0" $$

In this equation he equates the total virtual work of all the external forces to 0. This can only be true if external force is same as impressed force (non constraint force). Below is an example where this does not seem to be:

Consider a marble rolling inside a hemispherical bowl. The marble is our system. The force applied by the bowl on the marble is external to the system (marble) but is a constraint force. The constraint is that the marble has to be in touch with the bowl.

• What has gone wrong in this so called classic text? Please illuminate.

Answer 1

1. The system is supposed to consist of $n$ points at which the forces act. If no force acts at the $i$-th point, just set $F_i = 0$.

2. The assertion that "this can only be true if external forces are the impressed forces" is false (of course all constraint forces are ultimately external). It is precisely the content of d'Alembert's principle that one may omit the constraint forces in the principle of virtual work.

Answer 2

Explanation of doubt 1:

Consider a double pendulum made up of two massive rods. Here there are two degrees of freedom and hence two generalized coordinates. Assume that 5 separate non-constraint (applied) forces act on this system of double pendulum: ${\vec {F_1}}$ and ${\vec {F_2}}$ on the upper rod (rod 1) and ${\vec {F_3}}, {\vec {F_4}}$ and ${\vec {F_5}}$ on the lower rod (rod 2). Now ${\vec {F_1}}$ and ${\vec {F_2}}$ can be replaced (hypothetically) by a single force ${\vec {F_A}}$ which produces the same effect as ${\vec {F_1}}$ and ${\vec {F_2}}$ combined (as taught in undergrad physics courses). Similarly the three forces on rod 2 can be replaced by a single force ${\vec {F_B}}$. Hence, finally we have the number of forces just equal to the no. of degrees of freedom (no. of generalized coordiantes).

I can't think of a way of reducing the no. of forces further.

Explanation of doubt 2:

Let's hope that the author means 'impressed' or 'non-constraint' or 'applied' force when he says 'external' force.