The Cauchy problem in classical mechanics

Quantum mechanics is particularly reputed as being non-deterministic, but despite this, quite a large amount of theories, such as general relativity or electromagnetism can be non-deterministic. In particular, even a theory as mildly reputed as classical mechanics has the same issues, if we do not qualify properly what we mean by this.

Determinism, predictability and computability

There are three related notions which relate to these types of issues.

First there is determinism. Determinism concerns the theory (ie the theoretical terms) itself, and not how it relates to the real world.

Newtonian mechanics and determinism

Newtonian mechanics can be defined by a lot of things, but for now let's only consider it as the theory of point masses.

Newtonian mechanics happens on the Newtonian spacetime $\mathbb{R} \times \mathbb{R}^3$, equipped with the Euclidian metric $\delta$ on $\mathbb{R}^3$. On this, we have a (let's say) countable set of point masses $(x_i, m_i)$, $i \in \mathbb{N}$, each $x_i$ a map

\begin{eqnarray} x_i : \mathbb{R} &\to& \mathbb{R}^3\\ t &\mapsto& x_i(t) \end{eqnarray}

We can define them in terms of Lagrangian or Hamiltonian mechanics, but here the systems we'll consider are simple enough that we can simply stick to Newton's laws, and so each trajectory obeys the relation

\begin{eqnarray} m_i \ddot{x}_i(t) = F_i(x_j, t) \end{eqnarray}

We'll see how more requirements become necessary from the requirements of determinism as things go on. In particular, to help with some demonstrations, we won't require that the theory be covariant under the Galilean group.

The quintessential example of a Newtonian system is the free point mass, defined by

\begin{eqnarray} \ddot{x}_i(t) = 0 \end{eqnarray}

This is, due to the Picard–Lindelöf theorem, fairly easy to show to be deterministic. Given initial conditions $x_i(0) = x_i^0$ and $\dot{x}(0) = v_i^0$


Norton's dome

The most famous example of a non-deterministic system in Newtonian mechanics is Norton's dome, defined by

\begin{eqnarray} m \ddot{x}(t) = \sqrt{x(t)} \end{eqnarray}

The exact form of the problem varies. Typically, it involves an axially-symmetric surface of parametrization

\begin{eqnarray} z = -\frac{2}{3g} r^{\frac{3}{2}} \end{eqnarray}

So that its unit normal vector is

\begin{eqnarray} n = \frac{1}{1 + g^{-2} r} (1, \frac{1}{g} r^{\frac{1}{2}}) \end{eqnarray}

If we assume the simplified gravitational force $W = -mg e_z$, its components on the normal vector are

\begin{eqnarray} n \cdot W = -mg \end{eqnarray}

To have no movement along our normal direction, the reaction force $N = \alpha n$ has to be such that

\begin{eqnarray} (N + W) \cdot n &=& 0\\ &=& \alpha - mg &=& 0 \end{eqnarray}

So that $\alpha = mg$. Therefore, our

Space invaders and non-collision Painlevé singularities

A space invader corresponds to a particle "leaving" (or, in a similar manner, entering) Newtonian spacetime after a finite time. In other words, if we consider a coordinate system such that $x(0) = 0$, a space invader is such that, for $T \in \mathbb{R}$, we have

\begin{equation} \lim_{t \to T} |x(t)| = \infty \end{equation}

Our trajectory $x(t)$ is therefore unbounded. As it obeys Newton's laws, $x(t)$ must be at least $C^2$ on $[0,T)$. Since $|x(t)|$ is divergent, we have

\begin{equation} \forall X \in \mathbb{R}, \exists t_X < T, \forall t > t_X, |x(t)| > X \end{equation}

Consider the limit of our derivative,

\begin{equation} \lim_{t \to T} \frac{d}{dt} |x(t)| = \lim_{t \to T} \lim_{h \to 0} \frac{|x(t + h)| - |x(t)|}{h} \end{equation}


A fairly contrived but simple example is to consider a force unbounded on a finite interval, such as $F = \tan(t)$. To avoid any unpleasantness we can assume that this force is only defined on $(-\pi/2, \pi/2)$ and zero outside with an appropriate top hat function. Assuming unit mass (the mass isn't terribly important here), this is

\begin{equation} \ddot{x}(t) = \tan(t) \end{equation}

with solution

\begin{equation} \dot{x}(t) = \ln(\cos(t)) + v_0 \end{equation}

Systems of an infinite collection of point masses

Another way of getting

Boundary conditions for field theories

While technically not entirely a classical theory, electromagnetism can be and has been treated as so, simply as a classical wave theory in a medium (and indeed we could simply do study this instead if the idea of electromagnetism was too relativistic for us). Disregarding the whole gauge issue (as always, the issue will still be here in a different gauge), let's simply consider the vacuum Maxwell equation in the Lorenz gauge for simplicity.

\begin{equation} \Box A^\mu = 0 \end{equation}

To simplify things somewhat, we can also use simply the wave equation, either for computing any component of our electromagnetic potential, describing a Klein-Gordon field, or any mechanical wave system. Then we simply get

\begin{equation} \Box \phi = 0 \end{equation}

To solve this, we need the Green's function of the d'Alembert equation.

\begin{equation} G(t, x; t', x') = \frac{1}{|x - x'|}\left[ c_R \delta((t - t') + |x - x'|) + c_A \delta((t - t') - |x - x'|)\right] \end{equation}

with $c_R + c_A = 1$. The appearance of a free parameter in the Green's function seems like bad news, and indeed does not end well. Typically, we have that $c_R = 1$, $c_A = 0$, but if we take any other approach, then consider the solutions. Another way in which we try to constrain the set of solutions is by limiting ourselves to the set of Schwartz functions,

\begin{equation} \mathcal{S} = \end{equation}
Last updated : 2019-09-24 11:39:46
Tags : physics , classical-mechanics