5.2. Lagrange Equations of the First Kind#
Explicitly making appear constraint forces, due to constraints
\[\begin{split}\begin{aligned}
& \dfrac{d}{dt}\frac{\partial L}{\partial \dot{q}^k} - \frac{\partial L}{\partial q^k} = Q^e_k + Q^c_k \\
& g^j \left(q^k(t), t\right) = 0
\end{aligned}\end{split}\]
Example 5.1
Pendulum with point mass \(m\) and length \(\ell\), with hinge position \(x_H(t)\) w.r.t. an inertial reference frame, in a gravitational field \(\vec{g} = g \hat{y}\)
Position, and velocity of the point mass in \(P\)
\[\begin{split}\begin{aligned}
& \vec{r}_P(t) = x_P(t) \, \hat{x} + y_P(t) \, \hat{y} = \left( x_H(t) + \ell \sin \theta(t) \right) \, \hat{x} + \ell \cos \theta(t) \, \hat{y} \\
& \vec{v}_P(t) = \dot{x}_P(t) \, \hat{x} + \dot{y}_P(t) \, \hat{y} = (\dot{x}_H + \ell \dot{\theta}(t) \cos \theta(t)) \, \hat{x} - \ell \dot{\theta}(t) \sin \theta(t) \, \hat{y} \\
\end{aligned}\end{split}\]
Approach 1. LE of the II Kind. LE of the II Kind provides free equations of motion. The system has one degree of freedom. Here the angle \(\theta(t)\) is chosen as the generalized dof. Kinetic energy \(K\) and potential function \(U\),
\[\begin{split}\begin{aligned}
K & = \frac{1}{2} m \left( \dot{x}_H^2 + 2 \ell \dot{x}_H \dot \theta \cos \theta + \ell^2 \dot{\theta}^2 \right) \\
U & = m g \ell \cos \theta \\
\end{aligned}\end{split}\]
and Lagrange equation of the II-kind provides a free equation of motion, that immediately follows from direct evaluation of the required derivatives
\[\begin{split}\begin{aligned}
\dfrac{d}{dt}\frac{\partial L}{\partial \dot{\theta}}
& = \dfrac{d}{dt} \left[ m (\ell \dot{x}_H \cos \theta + \ell^2 \dot{\theta} ) \right]
= m \ell \ddot{x}_H \cos \theta - m \ell \dot{x}_H \dot{\theta} \sin \theta + m \ell^2 \ddot{\theta} \\
\frac{\partial L}{\partial \theta}
& = - m \ell \dot{x}_H \dot{\theta} \sin \theta - m g \ell \sin \theta
\end{aligned}\end{split}\]
Thus, Lagrange equation reads
\[
0 = \dfrac{d}{d t} \frac{\partial L}{\partial \dot{\theta}} - \frac{\partial L}{\partial \theta}
= m \ell^2 \ddot{\theta} + m g \ell \sin \theta - m \ell \ddot{x}_H (t) \cos \theta
\]
Approach 2. LE of the I Kind.