Classical Mechanics

Summary
  • We’ll talk about the two frameworks that can be used in classical mechanics and why one of them is much better than the other.
  • You’ll understand how we can derive Newton’s famous second law $ F = m \frac{d²}{dt²}q =m \ddot q.$
  • You’ll understand the historic roots of the most important idea of modern physics and why it is so important.

It’s hard to find good explanations for the core ideas of classical mechanics, because classical mechanics isn‘t as sexy as quantum mechanics and isn’t always correct.

It works great with big, everyday objects like rolling balls. but fails to describe nature at the most fundamental scale. We can‘t expect that an approximately correct theory is beautiful and many aspects of classical mechanics are ugly.

Okay, you may think, this isn‘t a big problem, because even if Mr. Dirac claims

“… it is more important to have beauty in one’s equations than to have them fit experiment”

there is no good reason why our physical theories should be simple and beautiful. Unfortunately some aspects of classical mechanics are not only ugly, but impossible to understand without knowledge of the correct theories, like special relativity.

That said, let’s take a look at the ugly stuff.

Newton’s Framework

The equations and the framework of classical mechanics were deduced historically from experiments. This worked pretty good, but is highly unsatisfactory from a theoretical point of view. Newton proposed

\begin{equation} \label{newtonssecond}  \tag{1} F = m \frac{d²}{dt²}q =m \ddot q,  \end{equation}

where $m$ is the mass, $\ddot q$ the acceleration and $F$ the force that acts on the object in question.

To describe some object we simply have to deduce equations for the forces $F$ that act on the object from experiments and put them on the left-hand side of the equation. This yields a differential equation, which we must solve for $q=q(t)$.

The solution is called the trajectory of the object and describes the position of the object for every moment in time. This is one framework for classical mechanics and it‘s useful for many, many things.

This is why every student of physics must solve Eq. \ref{newtonssecond} for many different situations.

In physics merely describing things is not enough. We want answers to the question: Why?. Why $ F = m \ddot q $ and not $ F = m \dddot q $ or $ F = m \dot q $? Is there any deeper explanation for the equations we deduced from experiments for the forces?

Happily, there is another framework for classical mechanics that enables us to give answers to these questions. At first sight it may look much more complicated, but you‘ll get used to it and learn to love it. This second framework is commonly called the Lagrangian formalism.

Lagrange‘s Framework

Joseph-Louis Lagrange, an Italian mathematician with a distinctive nose, was fascinated by something called Fermat’s principle.

This principle, discovered more than hundred years before Lagrange was born, states that „light travels between two given points along the path of shortest time”.

This is beautiful. This is how every theory of nature should work. No complicated formulas that we need to deduce from experiments and not just some law we can use, but don‘t understand.

If I could build a universe with physical laws from scratch, this is exactly what I would use as my starting point. Guess what?

Nature works this way! Every theory of physics that is in agreement with all experimental facts, like special relativity or quantum field theory is based on a similar principle. Why only similar and not the same?

Unfortunately a rolling ball does not simply obey Fermat‘s principle. Nonetheless Joseph-Louis Lagrange was confident that this idea is too beautiful to be wrong and started in his twenties to search for a similar principle that can be used to describe the trajectories of ordinary objects, like a rolling ball. Something that could substitute Newton’s somewhat arbitrary second law $ F = m \ddot q $.

Fermat‘s principle reads in mathematical terms: We need to minimize

$$ S[\mathbf{q}(t)]=\int_{\text{a}}^{\text{b}} dt . $$

The result of such a minimization procedure is the correct path $\mathbf{q}(t)$ light travels between the starting point $a$ and the end point $b$. Maybe you wonder how we can find the minimum of an integral, but that‘s a story for another day. For the moment believe me that it can be done.

After some time, Lagrange discovered that we get the correct trajectories if we minimize

$$ S[\mathbf{q}(t)]=\int_{\text{a}}^{\text{b}} (T-V) dt = \int_{\text{a}}^{\text{b}} L dt , $$

where $T$ is the kinetic energy, $V$ the potential energy and $ L= T-V$ is called the Lagrangian. Take note of the minus sign. If it were $(T+V)$ it would be simply the total energy, but that‘s not the case.

Admittedly, this does not look much better than Newton‘s $ F = m \ddot x. $ I remember beeing told by my professor in my first year at university that there is no explanation. One simply has to accept that we need to minimize strange function $L= T-V$.

That is so wrong. Why would we then talk about the Lagrange formalism nowadays?  Another ugly thing we can‘t explain? No thanks, one is enough.

There is a beautiful explanation, but first let’s ask some mathematician: What is the result if we minimize $S[\mathbf{q}(t)]$ ?

The result is the Euler-Lagrange equation:

$$\frac{\partial L(q,\dot{q},t)}{\partial q}- \frac{d}{d t}\left(\frac{\partial L(q,\dot{q},t)}{\partial \dot{q}}\right)=0 $$

To derive this equation one needs a new mathematical theory, called variational calculus, but unfortunately a full explanation would lead us too far apart from our main line of thought here.

Please accept this equation for the moment and see where it leads us. You can read about the derivation later. Let‘s put the Lagrangian $$L=T-V= \frac{1}{2}m \dot{q}^2 – V(q) ,$$

where we used the usual formula $T= \frac{1}{2}m \dot{q}^2 $ for the kinetic energy and assume, as usual in physics, that the potential energy $V$ depends only on the position and not the velocity, into the Euler-Lagrange equation:

\begin{align} & \quad \frac{\partial L}{\partial q}-\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}}\right)=0 \notag \\ &\rightarrow \frac{\partial }{\partial q} \left( \frac{1}{2}m \dot{q}^2- V(q) \right)-\frac{d}{d t}\left(\frac{\partial }{\partial \dot{q}} \left( \frac{1}{2}m \dot{q}^2 – V(q) \right) \right)=0 \notag \\ &\rightarrow – \frac{\partial }{\partial q} V(q) – \frac{d}{d t} m \dot{q} =0 \notag \\ &\rightarrow m \ddot{q} = – \frac{\partial }{\partial q} V(q)\end{align}

Looks familiar? This is exactly Newton’s $ F = m \ddot q $, if we recall the usual relationship between potential energy and a force $F= – \frac{\partial }{\partial q} V(q)$.

In fact, Lagrange made the general ansatz

$$S[\mathbf{q}(t)]= \int_{\text{a}}^{\text{b}} L dt $$

and then tested different functions $L$ until he found one that yields Newton’s law if put into the Euler-Lagrange equation. Apparently $ L=T-V$ is correct, but we still don‘t know why.

In classical mechanics there isn‘t any explanation, but Einstein‘s special relativity offers one. Again, diving into special relativity would be too much here, but let me show you something. In special relativity the Lagrangian is

$$ S[\mathbf{q}(t)]= -mc^2 \int_{\text{a}}^{\text{b}} d\tau $$

There is some constant $-mc^2$, where $m$ is the mass and $c$ the speed of light, but inside the integral we have something that looks like in Fermat‘s principle. There isn‘t some strange function $L$, just $ \int d\tau $.

Nature likes it simple.

One curious feature of special relativity is that there is no absolute time. Every observer measures its own time. Nevertheless, we can always imagine one observer that travels with the object in question. For this special observer the object in question would always be at rest, even it moves for all other observers.

This particular observer is interesting, because all other observers agree on the time this observer measures between the starting and the end point. The technical term for the time on the watch of this special observer is the proper time $\tau$.

Therefore in special relativity we minimize something plausible again. Classical mechanics is an approximation of special relativity if everything moves slowly compared with the speed of light. There is a mathematical tool, called the Taylor series, that enables us to inspect such limits and the result is that in this limit we must minimize exactly $T-V$.

This explains Newton’s second law and Lagrange’s strange function $L$. In the limit when everything moves slowly we get mysterious laws and functions. But if we take a look at the correct theory, instead of an approximation everything makes sense.

As if this weren‘t enough here a some more awesome facts about the Lagrangian formalism.

The Lagrangian Formalism is Used Everywhere in Physics. Here’s why:

  • The Lagrangian formalism is the best framework we have if we want to work with symmetries. If the corresponding Lagrangian does not change under some transformation, the transformation is a symmetry of the physical system. This is especially important in the search for new physical theories, because for example, one big condition is that the equations for our new theory are the same in all frames of reference. Otherwise we would have different equations for different observers and the theory would be useless. This translates directly to the restriction is that the Lagrangian must be invariant (which means does not change) under ALL transformation that leave the speed of light unchanged. This excludes a lot of theories and makes the search for the correct equations much simpler. For this reason the Lagrangian formalism is used everywhere in modern physics.
  • The Lagrangian formalism can be used to derive one of the most important theorems of modern physics: Noether‘s theorem. This theorem tells us that for every symmetry of our physical system we have a conserved quantity. For example, if our physical system is symmetric under rotations, we automatically know that angular momentum is conserved. This is an incredibly beautiful and deep insight.
  • The Lagrangian is directly connected to the total energy of the system in question by a transformation called Legendre transformation. The Legendre transformed Lagrangian is called the Hamiltonian of the system. The Hamiltonian plays a big role in quantum mechanics, because it tells us how things change in time.

Now, what are best books to learn more about this?

The Best Books about Classical Mechanics

  • As most students I love everything that was written by Feynman. Happily there is one big book by Feynman about Classical Mechanics: The Feynman Lectures on Physics Vol 1. Unfortunately, there isn’t much about the Lagrangian formalism there, but it’s great to learn about Newton’s approach.
  • To learn more about the Lagrangian formalism and the reason why it works this way I can’t recommend The Variational Principles of Mechanics by Lanczos enough.

More book recommendations can be found here.