Housekeeping

the syllabus and all course content are on my website, annaliw.github.io/teaching
email me any time with questions! This course has a lot of new information and you don’t want to get behind.
I’m on campus M/W/F. Stop by my office any time. You can make a zoom appointment T/Th.
This is my first time teaching such a small class, and my first time teaching an upper division physics course. I’m going to need some help from you to keep things going smoothly:
- help me understand what you do and don’t know. I’m still learning about the curriculum and program here.
- I like to run a socratic and discussion-based class. I need you to ask questions and participate! Let me know what you need from me to make this happen.
Homework is due at the beginning of class on Fridays, but I’ll accept it at my office until 1pm. After that I might be gone for the day. My goal is to grade the homework over the weekend and return it to you on Monday. That’s why there’s such a large late penalty.
Homework will be challenging. Get started early, form study groups, and come to office hours!

Classical mechanics

This semester we will rediscover classical mechanics from a new and much more powerful perspective.

equations of motion → phase space, generalized coordinates
forces drive motion → Lagrangian and Hamiltonian energy-based mechanics
circular orbits → general orbits around central force, including elliptical or hyperbolic
circular motion → general rotaion of rigid bodies along multiple axes
single simple harmonic oscillator → coupled and chaotic oscillators
inertial reference frames → non-inertial reference frames

We will cover a huge range of topics! All of them push into more advanced mathematical approaches, with greater generality and a view towards applications to other physics topics (quantum mechanics, general relativity, field theories).

Let’s start with a review of what we know from introductory mechanics:

Newton’s laws

A body remains at rest or in uniform motion unless acted on by an outside force.
A body acted on by a force moves in such a manner that the rate of change of motion equals the force. \(\vec F = m\vec a\).
If two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.

Although we will find that Newtonian mechanics has limits, we can still do a lot of work from these principles. Today we will warm up and discover how much more we can do with the math we have learned since intro, while staying in the framework of Newtonian mechanics.

Drag and air resistance}

Drag and air resistance are not problems that we solve analytically in intro mechanics. You may have done a numerical calculation. Now, with the power of differential equations, we can solve some of these problems.

Linear drag

Let’s first consider a linear drag force law, \(F=-c_1 v\), and start with Newton’s second law.

\[\begin{aligned} m\ddot{x} = m\dot{v} &= -c_1 v\end{aligned}\]

By writing this in terms of the first derivative of velocity, we have a simple, separable, ordinary differential equation.

\[\begin{aligned} m\dot{v} &= -c_1 v \\ \frac{1}{v}\dot{v} &= \frac{-c_1}{m} \\ \int_{v_0}^{v(t)} \frac{1}{v}dv &= \int_0^t \frac{-c_1}{m} dt \\ \log\left( \frac{v(t)}{v_0} \right) &= \frac{-c_1}{m} t \\ \frac{v(t)}{v_0} &= e^{\frac{-c_1}{m} t} \\ v(t) &= v_0 e^{\frac{-c_1}{m} t} \end{aligned}\]

To find an equation for position, we just integrate again.

\[\begin{aligned} v(t) = \frac{dx}{dt} &= v_0 e^{\frac{-c_1}{m} t} \\ \int_0^{x(t)} dx &= v_0 \int_0^t e^{\frac{-c_1}{m} t} \\ x(t) &= \frac{-mv_0}{c_1} \left( e^{\frac{-c_1}{m} t} - 1 \right) \\ &= \frac{mv_0}{c_1} \left( 1- e^{\frac{-c_1}{m} t} \right) \end{aligned}\]

Notice that as \(t\rightarrow\infty\) the velocity eventually goes to zero and the position approaches a limiting value \(\frac{mv_0}{c_1}\).

Quadratic drag

We can follow a similar procedure for a quadratic drag force (like air resistance), \(F = -c_2v^2\). \(\begin{aligned} m\ddot{x} = m\dot{v} &= -c_2 v^2 \\ \frac{1}{v^2}\dot{v} &= \frac{-c_2}{m} \\ \int_{v_0}^{v(t)} \frac{1}{v^2} dv &= \int_0^t \frac{-c_2}{m} dt \\ -\frac{1}{v(t)} + \frac{1}{v_0} &= \frac{-c_2}{m} t \\ v(t) &= \left( \frac{1}{v_0} + \frac{c_2t}{m} \right)^{-1} \\ &= \left( \frac{m + v_0c_2t}{mv_0} \right)^{-1} \\ &= \frac{mv_0}{m + v_0c_dt} = \frac{v_0}{1+\frac{v_0c_d}{m}t} \\ v(t) &= \frac{v_0}{1 + kt} \end{aligned}\)

In the last step, I have set \(k = \frac{v_0c_d}{m}\) to simplify our notation. We can integrate again to get position.

\[\begin{aligned} v(t) = \frac{dx}{dt} &= \frac{v_0}{1 + kt} \\ \int_0^{x(t)} dx &= \int_0^t \frac{v_0}{1 + kt} dt \\ u = kt \\ x(t) &= \int_0^u \frac{v_0}{k} \frac{du}{1+u} \\ v = 1+u \\ x(t) &= \int_1^{1+u} \frac{v_0}{k} \frac{dv}{v} \\ &= \frac{v_0}{k} \left( \log(1 + u) - \log(1) \right) \\ &= \frac{v_0}{k} \log(1+kt) \end{aligned}\]

For this result, \(x(t)\) does not reach an asymptotic value but instead continues to grow very slowly.

This is a fully Newtonian result, but much more advanced than what you were able to accomplish in your intro mechanics class. Next time we will explore some areas where Newtonian mechanics does not hold up.