SICM Exercise 1.09
Solution to exercise 1.9 of Structure and Interpretation of Classical Mechanics by Gerald Jay Sussman and Jack Wisdom.
Lagrange’s equations
🛈 Note
Derive the Lagrange equations for the following systems, showing all of the intermediate steps as in the harmonic oscillator and orbital motion examples.
- An ideal planar pendulum consists of a bob of mass \(m\) connected to a pivot by a massless rod of length \(l\) subject to uniform gravitational acceleration \(g\). A Lagrangian is \(L(t, \theta, \dot{\theta})) = \frac{1}{2} m l^2 \dot{\theta}^2 + m g l \cos \theta\). The formal parameters of \(L\) are \(t\), \(\theta\), and \(\dot{\theta}\); \(\theta\) measures the angle of the pendulum rod to a plumb line and \(\dot{\theta}\) is the angular velocity of the rod.
- A particle of mass m moves in a two-dimensional potential \(V(x, y) = (x^2 + y^2)/2 + x^2y − y^3/3\), where \(x\) and \(y\) are rectangular coordinates of the particle. A Lagrangian is \(L(t;x,y;v_x,v_y)=\frac{1}{2}m(v^2_x+v^2_y) − V(x,y)\).
- A Lagrangian for a particle of mass \(m\) constrained to move on a sphere of radius \(R\) is \(L(t; \theta, \varphi; \alpha, \beta) = \frac{1}{2} m R^2 (\alpha^2 + (\beta \sin \theta)^2)\). The angle \(\theta\) is the colatitude of the particle and \(\varphi\) is the longitude; the rate of change of the colatitude is \(\alpha\) and the rate of change of the longitude is \(\beta\).
Ideal planar pendulum
The Lagrangian for the ideal planar pendulum is
\[L(t, \theta, \dot{\theta}) = \frac{1}{2} m l^2 \dot{\theta}^2 + m g l \cos{\theta}\]The partial derviatives with respect to the second and third parameters are
\[\partial_1 L(t, \theta, \dot{\theta}) = -m g l \sin{\theta}\]\[\partial_2 L(t, \theta, \dot{\theta}) = m l^2 \dot{\theta}\]
Substituting the time dependent path function \(\Gamma[\theta](t) = (t, \theta (t), D \theta (t))\)
\[(\partial_1 L \circ \Gamma [\theta])(t) = -m g l \sin{\theta(t)}\]\[(\partial_2 L \circ \Gamma [\theta])(t) = m l^2 D\theta\]
The derivative of the partial differential equation with respect to the second parameter is
\[D(\partial_2 L \circ \Gamma [\theta])(t) = m l^2 D^2\theta\]Assembling the Lagrange equation
\[m l^2 D^2 \theta(t) + m g l \sin(\theta(t)) = 0\]We can simplify with \(g l\)
\[l D^2 \theta(t) + g \sin(\theta(t)) = 0\]Particle of mass \(m\)
The potential function is
\[V(x, y) = \frac{x^2 + y^2}{2} + x^2 y - \frac{y^3}{3}\]The Lagrangian of the system is
\[L(t; x, y; v_x, v_y) = \frac{1}{2} m (v_x^2 + v_y^2) - V(x, y)\]The partial derviatives with respect to the second and third parameters are
\[\partial_1 L(t; x, y; v_x, v_y) = [\partial_{1,0} L(...), \partial_{1,1} L(...)] = [-x - 2xy, -y - x^2 + y^2]\]\[\partial_2 L(t; x, y; v_x, v_y) = [\partial_{2,0} L(...), \partial_{2,1} L(...)] = [mv_x, mv_y]\]
Substituting the time dependent path function \(\Gamma[q](t) = (t, x(t), y(t), Dx(t), Dy(t))\)
\[(\partial_1 L \circ \Gamma [q])(t) = [-x(t) - 2 x(t) y(t), -y(t) - x^2(t) + y^2(t)]\]\[(\partial_2 L \circ \Gamma [q])(t) = [m Dx(t), m Dy(t)]\]
The derivative of the partial differential equation with respect to the second parameter is
\[D(\partial_2 L \circ \Gamma [q])(t) = [m D^2x(t), m D^2y(t)]\]The Lagrange equations are
\[m D^2 x(t) + x(t) + 2 x(t) y(t) = 0\]\[m D^2 y(t) + y(t) + x^2(t) - y^2(t) = 0\]
Particle of mass \(m\) moving on a sphere of radius \(R\)
The Lagrangian of this system is
\[ L(t; \theta, \varphi; \alpha, \beta) = \frac{1}{2} m R^2 \left( \alpha^2 + {\left( \beta \sin \theta \right) }^2 \right)\]The partial derviatives with respect to the second and third parameters are
\[ \partial_1 L(t; \theta, \varphi; \alpha, \beta) = \left[\partial_{1,0} L(...), \partial_{1,1} L(...)\right] = \left[m R^2 \beta^2 \sin\theta \cdot \cos\theta, 0 \right] \]\[ \partial_2 L(t; \theta, \varphi; \alpha, \beta) = \left[\partial_{2,0} L(...), \partial_{2,1} L(...)\right] = \left[m R^2 \alpha, m R^2 \beta \sin^2(\theta)\right] \]
Substituting the time dependent path function \( \Gamma[q](t) = (t, \theta(t), \varphi(t), D\theta(t), D\varphi(t)) \)
\[ \partial_1 L \circ \Gamma[q] = \left[m R^2 \sin\theta(t)\cos\theta(t) (D\varphi(t))^2, 0 \right] \]\[ \partial_2 L \circ \Gamma[q] = \left[m R^2 D\theta(t), m R^2 \sin^2\theta(t) D\varphi(t) \right] \]
The derivative of the partial differential equation with respect to the second parameter is
\[ D(\partial_2 L \circ \Gamma[q]) = \left[m R^2 D^2\theta(t), m R^2 \left(2\sin\theta(t) \cos\theta(t) D\theta(t) D\varphi(t) + \sin^2\theta(t) D^2\varphi\right) \right] \]The Lagrange equations are
\[D^2\theta(t) - \sin\theta(t)\cos\theta(t) (D\varphi(t))^2 = 0\]\[2\sin\theta(t) \cos\theta(t) D\theta(t) D\varphi(t) + \sin^2\theta(t) D^2\varphi = 0\]