Classical Mechanics (Goldstein) · Problem 4

Survey of the Elementary Principles — Herbert Goldstein, Charles Poole, John Safko

Physics — Mechanics · nonholonomic-constraints-integrating-factor

Problem

The equations of constraint for the rolling disk, Eqs. (1.39), are special cases of general linear differential equations of constraint of the form

\sum_{i=1}^{n} g_i(x_1, \ldots, x_n) dx_i = 0.

A constraint condition of this type is holonomic only if an integrating function

f(x_1, \ldots, x_n)

can be found that turns it into an exact differential. Clearly the function must be such that

\frac{\partial (f g_i)}{\partial x_j} = \frac{\partial (f g_j)}{\partial x_i}

for all

i \neq j

. Show that no such integrating factor can be found for either of Eqs. (1.39).