Question 11.7: A nonconservative holonomic system having two degrees of fre...

Notice that  q_2  is ignorable and  Q_2^N=0.  Therefore, we have  immediately by (11.47) the corresponding momentum integral

p_e\left(\dot{q}_r, q_r, t\right)=\frac{\partial L\left(\dot{q}_r, q_r, t\right)}{\partial \dot{q}_e}=\gamma_e, \text { a constant. }                                (11.47)

p_2=\frac{\partial L}{\partial \dot{q}_2}=2 m b^2\left(\dot{q}_2  +  \frac{a}{b} \cos q_1\right)=\gamma_2, \text { a constant. }                          (11.48b)

Caution:We must continue to apply the Lagrange equations to  (11.48a) in which all of the variables are considered independent. Equation (11.48b) is a partial solution of one of these equations that necessarily relates these variables; but it is not to be substituted into the Lagrangian, it is to be used in connection with the companion equation for  q_1.  The second of Lagrange’s equations (11.38) yields

m b^2 \ddot{q}_1  +  2 m a b\left(\dot{q}_2  +  \frac{a}{b} \cos q_1\right) \sin q_1  –  m a^2 \sin q_1 \cos q_1=Q_1^N=-m b^2 v \dot{q}_1 .                          (11.48c)

We now use (11.48b) to eliminate  \dot{q}_2  from (11.48c) to obtain

\ddot{q}_1  +  v \dot{q}_1  +  \frac{a}{b} \sin q_1\left(\frac{\gamma_2}{m b^2}  –  \frac{a}{b} \cos q_1\right)=0                            (11.48d)

The two equations (11.48b) and (11.48d), in principle, determine  q_1(t)  and  q_2(t).