Chapter 3

In Example $3.8$, find ${\tilde{Q}}_1$ and ${\tilde{Q}}_2$ in terms of the components of $\mathit{F}$ by considering the work done under suitable small displacements. Check that the same expressions for the ${\tilde{Q}}_a$ s follow from the chain rule in the case that $\mathit{F}=-\mathbf{\nabla }U$.

A particle of mass $m$ is subject to a force $\mathit{F}$. Obtain the equations of motion in cylindrical polar coordinates.

A particle of unit mass is subject to an inverse-square-law central force $$\mathit{F}=-\frac{\mathit{r}}{r^3}$$ where $r=|\mathit{r}|$ and $\mathit{r}$ is the position vector from the origin of an inertial frame. Show that the motion is governed by the Lagrangian $$L=\frac{1}{2}\dot{r}\cdot \dot{r}+\frac{1}{r}$$ Write down the equations of motion in spherical polar coordinates and show that there are solutions with $\theta =\pi /2$ throughout the motion.

Two particles, each of mass $m$, are moving under their mutual gravitational attraction, which is given by the potential $U=-\gamma m/2r$, where $2r$ is their separation and $\gamma$ is a constant. Find the equations of motion in terms of the coordinates $X,Y,Z,r,\theta ,\phi$, where $X,Y$, and $Z$ are the Cartesian coordinates of the centre of mass and $r,\theta$, and $\phi$ are the polar coordinates of one particle relative to the centre of mass.

The dynamics of a system with $n$ degrees of freedom are governed by a Lagrangian $L(q,v,t)$. Show that if $f(q,t)$ is any function on $CT$, then $$L^{\prime }=L+\frac{\partial f}{\partial q_a}{v}_a+\frac{\partial f}{\partial t}$$ generates the same dynamics.

A system has Lagrangian $L=\frac{1}{2}{T}_{ab}{v}_a{v}_b$ where the $T_{ab}\mathrm{s}$ are functions of the $q_a\mathrm{s}$ alone. Show that $$\frac{\mathrm{d}L}{\mathrm{d}t}=0$$ Show that if $f$ is any function of one variable, then $L^{\prime }=f(L)$ generates the same dynamics.

${}^{†}$ 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 $U(\mathit{r})$, where $\mathit{r}$ 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 $$V=\frac{m_1+m_2}{m_1}U.$$

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 $$x=\cosh \phi \cos \theta ,y=\sinh \phi \sin \theta ,$$ then the Lagrangian governing the motion becomes $$L=\frac{1}{2}\mathrm{\Omega }({\dot{\theta }}^2+{\dot{\phi }}^2)+2{\mathrm{\Omega }}^{-1}\cosh \phi ,$$ where $\mathrm{\Omega }={\cosh }^2\phi -{\cos }^2\theta$. Write down the equations of motion and show that if the particle is set in motion with $T+U=0$, then $$\frac{1}{2}({\theta }^2+{\dot{\phi }}^2)-\frac{2\cosh \phi }{{\mathrm{\Omega }}^2}=0$$ Deduce that $$\frac{{\mathrm{d}}^2\theta }{\mathrm{d}{\tau }^2}=0,\frac{{\mathrm{d}}^2\phi }{\mathrm{d}{\tau }^2}=2\sinh \phi$$ where $\mathrm{d}t/\mathrm{d}\tau =\mathrm{\Omega }$. Hence find the path of the particle.

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 $$L=\frac{1}{2}m({\dot{r}}^2+r^2{\dot{\theta }}^2)-\frac{\lambda }{2a}(r-a{)}^2$$ where $r$ and $\theta$ are plane polar coordinates with origin $O$.

A particle of mass $m$ is constrained to move under gravity on the surface of a smooth right circular cone of semi-vertical angle $\pi /4$. The axis of the cone is vertical, with the vertex downwards. Find the equations of motion in terms of $z$ (the height above the vertex) and $\theta$ (the angular coordinate around the circular cross-sections). Show that $${\dot{z}}^2+\frac{h^2}{2z^2}+gz=E$$ where $E$ and $h$ are constant. Sketch and interpret the trajectories in the $z,\dot{z}$-plane for a fixed value of $h$.