A nonconservative holonomic system having two degrees of freedom with generalized coordinates $\left(q_1, q_2\right)$ and corresponding generalized forces $Q_1^N=-m b^2 v \dot{q}_1, Q_2^N=0$ , has a Lagrangian function

$L=\frac{1}{2} m a^2 \sin ^2 q_1 + m b^2\left(\dot{q}_2 + \frac{a}{b} \cos q_1\right)^2 + \frac{1}{2} m b^2\left(\dot{q}_1 + c\right)^2,$ (11.48a)

in which a, b, c, and m are constants. Derive the Lagrange equations.

Question

A nonconservative holonomic system having two degrees of freedom with generalized coordinates [latex]\left(q_1, q_2\right)[/latex] and corresponding generalized forces [latex]Q_1^N=-m b^2 v \dot{q}_1, Q_2^N=0[/latex], has a Lagrangian function

[latex]L=\frac{1}{2} m a^2 \sin ^2 q_1 + m b^2\left(\dot{q}_2 + \frac{a}{b} \cos q_1\right)^2 + \frac{1}{2} m b^2\left(\dot{q}_1 + c\right)^2,[/latex] (11.48a)

in which a, b, c, and m are constants. Derive the Lagrange equations.

Accepted Answer

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

[latex]p_e\left(\dot{q}_r, q_r, t ight)=\frac{\partial L\left(\dot{q}_r, q_r, t ight)}{\partial \dot{q}_e}=\gamma_e, ext { a constant. }[/latex] (11.47)

[latex]p_2=\frac{\partial L}{\partial \dot{q}_2}=2 m b^2\left(\dot{q}_2 + \frac{a}{b} \cos q_1 ight)=\gamma_2, ext { a constant. }[/latex] (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 [latex]q_1[/latex]. The second of Lagrange’s equations (11.38) yields

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

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

[latex]\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 ight)=0[/latex] (11.48d)

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

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

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