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I am self-studying Goldstein's book "Classical Mechanics", and I need some help understanding the part where Goldstein discusses using Hamilton's principle to solve systems with holonomic constraints (Section 2.4). He writes on pg 46 (International Edition):

First consider holonomic constraints. When we derive Lagrange's equation from either Hamilton's or D'Alembert's principle, the holonomic constraint appear in the last step when the variations in the $q_i$ were considered independent of each other. However, the virtual displacements in the $\delta q_{I}$'s may not be consistent with constraints. If there are $n$ variables and $m$ constraint equations $f_\alpha$ of the form Eq. (1.37), the extra virtual displacements are eliminated by the method of Lagrange undetermined multipliers.

I do not understand the parts that the virtual displacements may not be inconsistent with constraints because earlier on in the book he defines virtual displacement as the infinitesimal change of the coordinates consistent with the forces and constraints imposed on the system at the given instant $t$ (pg 16).

What am I missing?

Qmechanic
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mononono
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2 Answers2

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Consider a bead sliding on a thin, rigid rod in the $x$-direction. An example of virtual infinitesimal displacement that is inconsistent with the constraints is $$ \delta \mathbf r = (0, \delta y, 0), \qquad \delta y \neq 0 $$ because this displacement represents the bead moving away from the rod.

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joshphysics
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I understand this comment by Goldstein as a motivation for Lagrangian multipliers.

In fact virtual displacements have to be, by definition, consistent with the constraints. Now let us consider a system with $N$ generalized coordinates $q_k$ and let us apply the Hamilton's principle. In order to get the Euler-Lagrange equations we need to consider that all the variations $\delta q_k$ are arbitrary and independent of each other. Suppose we have one constraint, $f(q_1,\ldots,q_N)=0$. This means that one of the coordinates, say $q_N$, is no longer independent and its variation $\delta q_N$ cannot be arbitrary, it has to be consistent with constraint. In order to obtain the Euler-Lagrange equation for the $q_N$ coordinate we must use the method of Lagrangian multipliers. Of course we could avoid this if we have started with $N-1$ independent generalized coordinates.

Moreover, the expression the virtual displacements in the $\delta q_k$ may not be consistent with constraints may not be the best. I suppose that by virtual displacement he really means virtual change (in the sense of Cornelius Lanczos' book, i.e. an infinitesimal change in $q_k(t)$ that does not come from an infinitesimal change in its argument but comes from an arbitrary change of the type $\epsilon\phi(t)$).

Diracology
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