Newtonian Mechanics

Intuitive

Newtonian mechanics is the standard theory that allows us to describe every day object like an apple that falls to the ground.

We do this by solving, so called equation of motion. The solutions of these equations are particle paths which describe how a particle will move as time passes on.

For example, the path of a point-like object could look as follows:

The basis of Newtonian Mechanics is summarized by three laws, commonly called "Newton's laws of motion":

  • First law: No force is needed to keep an object moving. If an object is at rest, it will remain at rest unless a force acts on it. Similarly, if an object moves with some constant velocity, it will keep moving unless a force acts on it.
  • Second law: The way the movement of an object changes depends only on two things: its mass and the total force acting on it.
  • Third law: Whenever an object exerts a force on another object, inevitably this second object will also exert a force of equal magnitude on the first object.

Concrete

The equations and the theory of Newtonian mechanics were deduced historically from experiments. This worked pretty good but is highly unsatisfactory from a theoretical point of view. Newton proposed his second law

\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.

Every student of physics has to solve Newton's second law for many different situations.

First law:

If the forces acting on an object are balanced, i.e. the total force is zero $\vec F=0$, the velocity of the object will remain constant: $\vec v=\text{const}$. So when the velocity is zero, it will remain zero. If the velocity has some other value it will keep it.

Forces are balanced $ \vec F=0$
$ \vec a=0$
Object at rest: $\vec v=0$ Object in motion $ \vec v\neq 0$
Object stays at rest: $\vec v=0$ Object remains in motion $ \vec v \neq 0$; same $ \vec v$.

Second law:

$$ \vec F = m \vec a$$

Forces are unbalanced $ \vec F\neq 0$
$ \vec a \neq 0$
acceleration $\vec a$ depends directly on the net-force $\vec F$ that acts on the object acceleration $\vec a$ depends inversly on the mass $m$ of the object

Recommended Resources

Abstract

Newton's second law is \begin{equation} F=ma \end{equation} which describes a particle moving in $\mathbb{R}^n$.

Its position, which we call $q$, depends on the time $t\in\mathbb{R}$.

Therefore, the position defines a function, \[ q \colon \mathbb{R}\longrightarrow\mathbb{R}^n. \] Using this function $q$ we can define the corresponding velocity, \[ v=\dot{q} \colon \mathbb{R}\longrightarrow\mathbb{R}^n \] where $\dot{q}=\frac{dq}{dt}$. Analogously, we can define the acceleration \[ a=\ddot{q} \colon \mathbb{R}\longrightarrow\mathbb{R}^n. \] We call $m>0$ be the mass of the particle. The last puzzle piece in the equation $F$ is a vector field on $\mathbb{R}^n$ which we call the force. Newton second law is 2nd-order differential equation for $q\colon\mathbb{R}\rightarrow\mathbb{R}^n$. It has a unique solution given some $q(t_0)$ and $\dot{q}(t_0)$, provided the vector field $F$ is smooth and bounded (i.e., $|F(x)|<B$ for some $B>0$, for all $x\in\mathbb{R}^n$).

  • A great introduction to high-level concepts in classical mechanics are the lecture notes by David Tong: http://www.damtp.cam.ac.uk/user/tong/dynamics.html Especially part 4 is amazing and explains, for example, nicely how similar classical and quantum mechanics are, if formulated in the same language. (E.g. in classical mechanics, the generators of translations are also given by the momentum, acting via the Poisson bracket.)
  • A great book that describes the high-level perspective on classical mechanics is "Introduction to Mechanics and Symmetry" by Jerrold E. Marsden and Tudor S. Ratiu and see also the other books by Marsden, many of which are freely available online here, for example, his "Foundations of Mechanics".

Why is it interesting?

Classical Mechanics is the framework that we use to describe the behaviour of objects we meet in everyday life: from the famous apple that falls from a tree, to the motion of planets in our solar system.

In addition to these "school physics" application, there is a high relevance to almost all topics in modern physics. Namely, it turns out that there is not one unique way to describe classical mechanics but several. Instead, we can describe classical mechanics using:

  • Newtonian Mechanics
  • Lagrangian Mechanics
  • Hamiltonian Mechanics

Each approach has unique strengths and it usually depends on the system that we want to describe which approach is best suited. It is important to note that these three approaches are completely equivalent in what they do. This means, the physics they describe is exactly the same. However, dependending on the system in question, sometimes calculations and the interpretation is easier in Lagrangian Mechanics and sometimes in Hamiltonian Mechanics.

It makes sense to study these various approaches instead of just one because:

  1. In more modern theories like quantum mechanics or quantum field theory, we are in exactly the same situation: There are several descriptions that all do the same thing. But for specific situations one approach is stronger than the others. To get a full understanding of the theory it is helpful to know all the various approaches. It is thus sensible to get familiar with this "various approaches" idea.
  2. The same three approaches are also applicable in quantum field theory and quantum mechanics, which are the best theories of nature that we currently have. In classical mechanics we deal with objects that we are familiar with in everyday life, whereas in quantum field theory and quantum mechanics, things are more abstract. Thus it makes sense to get familiar with the approaches using everyday objects.

Moreover, classical mechanics is an ideal playground to get familiar with many important mathematical tools and concepts.

Nature and Nature’s laws lay hid in night:

God said, "Let Newton be!" and all was light. Alexander Pope

FAQ

Why don't we use operators in classical mechanics, like we do in quantum mechanics?
Actually, we can do use operators:

As an interesting curiosity, classical mechanics can be formulated using wavefunctions/kets on a complex Hilbert space, with physical observables represented by operators and measurement being probabilistic, just as in quantum mechanics. It's not even all that strange, turning into a complexified reformulation of the classical Liouville equation (which handles probability distributions over the phase space).

The common formalism of QM is quite general and is completely capable of handling classical mechanics; where they disagree is on which operators represent things we actually measure in the world.

https://physics.stackexchange.com/a/80822/37286

See also: http://en.wikipedia.org/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics

What group belongs to the Lie algebra of classical mechanics?
The Lie algebra of classical mechanics is defined through the Poisson bracket. The corresponding Lie group which integrates the Poisson bracket is called the “quantomorphism group”. (Source: https://physics.stackexchange.com/a/75775/37286 )

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