Problem 1.

Consider a mass $m$ moving on a circular orbit under the influence of a central force of potential $V(r)$. Study small oscillations around the circular orbit, i.e. the motion of the particle when it is slightly deflected out of the equilibrium trajectory. Consider oscillations occurring in the same plane of the circular orbit and with the same angular momentum.
a) We perturb the path to linear order around the circular orbit,
$$r(t)=r_0+\epsilon \rho(t)+\mathcal{O}\left(\epsilon^2\right), \quad \varphi(t)=\omega t+\epsilon \theta(t)+\mathcal{O}\left(\epsilon^2\right)$$
where $r_0$ is the radius of the circular orbit, $\omega$ is the equilibrium frequency and $\epsilon \ll 1$. Determine the equations of motion for $\rho(t)$ and $\theta(t)$ at leading order in $\epsilon$.
b) Let the potential be of the form $V(r)=-k r^{-\alpha+1}$. Show that stable oscillations occur if $\alpha<3$.
Hint: Determine a relation between $\omega^2$ and $V^{\prime}\left(r_0\right)$ and use it.

Problem 2.

We consider a system of two identical pendula of length $l_1=l_2=l$ and masses $m_1=m_2=$ $m$ in a homogeneous gravitational field with acceleration $g$. The two pendula are moving in the same plane and we denote the (small) deflection angles by $\theta_1$ and $\theta_2$. Moreover, the pendula are connected by a massless spring, whose length equals the distance of the points to which the pendula are attached. We define $\omega_{\mathrm{g}}^2=g / l$ and $\omega_{\mathrm{s}}^2=k / m$.
a) Find the equations of motion for $\theta_1, \theta_2$ and the normal modes of the system.
b) At time $t=0$, the two pendula are at rest. Then, we push one of them such that it has the initial velocity $l \dot{\theta}1=v$. Show that the first pendulum is almost at rest after a time $T$, which you should determine. Assume that $\omega{\mathrm{s}} \ll \omega_{\text {g }}$.
Hint: $\sin a+\sin b=2 \cos \frac{1}{2}(a-b) \sin \frac{1}{2}(a+b)$.

Outline:

This is an introductory module in Classical Mechanics.  Starting from Newton’s Laws of Motion, it sets up the techniques used to apply the laws to the solution of physical problems.  It is essential background for many of the succeeding modules within the degrees in Physics and Astronomy. This module aims to allow students to understand the importance of classical mechanics in formulating and solving problems in many different areas of physics, develop problem-solving skills more generally, to introduce the basic concepts of classical mechanics and apply them to a variety of problems associated with the motion of single particles, interactions between particles and the motion of rigid bodies.

Aims:

This module aims to:
•    convey the importance of classical mechanics in formulating and solving problems in many different areas of physics and develop problem-solving skills more generally;
•    introduce the basic concepts of classical mechanics and apply them to a variety of problems associated with the motion of single particles, interactions between particles and the motion of rigid bodies.

Intended Learning Outcomes:

After completing this module students should be able to:

• state and apply Newton’s laws of motion for a point particle in one, two and three dimensions;
• use the conservation of kinetic plus potential energies to describe simple systems and evaluate the potential energy for a conservative force;
• understand an impulse and apply the principle of conservation of momentum to the motion of an isolated system of two or more point particles;
• evaluate kinematic quantities in the centre of mass system;
• solve for the motion of a particle in a one-dimensional harmonic oscillator potential with damping and understand the concept of resonance in a mechanical system;
• appreciate the distinction between inertial and non-inertial frames of reference, and use the concept of fictitious forces as a convenient means of solving problems in non-inertial frames;
• describe the motion of a particle relative to the surface of the rotating Earth through the use of the fictitious centrifugal and Coriolis forces;
• derive the conservation of angular momentum for an isolated particle and apply the rotational equations of motion for external torques;
• solve for the motion of a particle in a central force, in particular that of an inverse square law and so be able to describe planetary motion;
• describe the motion of rigid bodies, particularly when constrained to rotate about a fixed axis or when free to rotate about an axis through the centre of mass and and the motion of rolling objects with and without slipping;
• calculate the moments of inertia of simple rigid bodies and use the parallel and perpendicular axes theorems;
• appreciate the influence of external torques on a rotating rigid body.

Fourier analysis代写

## 离散数学代写

Partial Differential Equations代写可以参考一份偏微分方程midterm答案解析