Chapter 3: Problem 2
A particle of mass \(m\) is subject to a force \(\boldsymbol{F}\). Obtain the equations of motion in cylindrical polar coordinates.
Chapter 3: Problem 2
A particle of mass \(m\) is subject to a force \(\boldsymbol{F}\). Obtain the equations of motion in cylindrical polar coordinates.
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Get started for free\({ }^{\dagger}\) Two particles \(P_{1}\) and \(P_{2}\) have respective masses \(m_{1}\) and \(m_{2}\) and are attracted to each other by a force with time- independent potential \(U(\boldsymbol{r})\), where \(\boldsymbol{r}\) is the vector from \(P_{1}\) to \(P_{2}\). (1) Show that the motion of the centre of mass is the same as in the case \(U=0\). (2) Show that the motion of \(P_{2}\) relative to \(P_{1}\) is the same as in the case that \(P_{1}\) is fixed and \(P_{2}\) is attracted to \(P_{1}\) by a force with potential $$ V=\frac{m_{1}+m_{2}}{m_{1}} U . $$
Two particles, each of mass \(m\), are moving under their mutual gravitational attraction, which is given by the potential \(U=-\gamma m / 2 r\), where \(2 r\) is their separation and \(\gamma\) is a constant. Find the equations of motion in terms of the coordinates \(X, Y, Z, r, \theta, \varphi\), where \(X, Y\), and \(Z\) are the Cartesian coordinates of the centre of mass and \(r, \theta\), and \(\varphi\) are the polar coordinates of one particle relative to the centre of mass.
A particle of mass \(m\) is free to move in a horizontal plane. It is attached to a fixed point \(O\) by a light elastic string, of natural length \(a\) and modulus \(\lambda\). Show that the tension in the string is conservative and show that the Lagrangian for the motion when \(r>a\) is $$ L=\frac{1}{2} m\left(\dot{r}^{2}+r^{2} \dot{\theta}^{2}\right)-\frac{\lambda}{2 a}(r-a)^{2} $$ where \(r\) and \(\theta\) are plane polar coordinates with origin \(O\).
A particle of unit mass is moving in the \(x, y\) plane under the influence of the potential $$ U=-\frac{1}{r_{1}}-\frac{1}{r_{2}}, $$ where \(r_{1}\) and \(r_{2}\) are the respective distances from the points \((1,0)\) and \((-1,0)\). Show that if new coordinates are introduced by putting $$ x=\cosh \varphi \cos \theta, \quad y=\sinh \varphi \sin \theta, $$ then the Lagrangian governing the motion becomes $$ L=\frac{1}{2} \Omega\left(\dot{\theta}^{2}+\dot{\varphi}^{2}\right)+2 \Omega^{-1} \cosh \varphi, $$ where \(\Omega=\cosh ^{2} \varphi-\cos ^{2} \theta\). Write down the equations of motion and show that if the particle is set in motion with \(T+U=0\), then $$ \frac{1}{2}\left(\theta^{2}+\dot{\varphi}^{2}\right)-\frac{2 \cosh \varphi}{\Omega^{2}}=0 $$ Deduce that $$ \frac{\mathrm{d}^{2} \theta}{\mathrm{d} \tau^{2}}=0, \quad \frac{\mathrm{d}^{2} \varphi}{\mathrm{d} \tau^{2}}=2 \sinh \varphi $$ where \(\mathrm{d} t / \mathrm{d} \tau=\Omega\). Hence find the path of the particle.
A particle of unit mass is constrained to move on the surface of a unit sphere, but is otherwise free. Show that the dynamical trajectories are great circles traversed at uniform speeds. Show that if \(\gamma\) is a complete circuit of a great circle in time \(t\), then $$ J_{L}(v)=2 \pi^{2} / t $$ Does \(\gamma\) minimize \(J_{L}\) over all curves on the sphere that start and end at a point \(P\) on the equator and take time \(t\) for the round trip from \(P\) back to \(P ?\)
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