The classical view of the universe

In the late 19th and early 20th centuries, the science of physics and the prevailing view of the universe changed in a truly profound manner. The change was instigated by a series of key experiments that could not be rationalized within the existing theory of classical mechanics as laid out by Newton.

Let us recall what classical mechanics implies as a world view. A classical system consisting of $N$ particles is completely determined by specifying the positions and velocities of all particles at any point in time. See the figure below:

Figure 1:

Since momentum ${\bf p}= m{\bf v}$, we can also specify each particle's positions ${\bf r}$ and its momentum ${\bf p}$. When this is done for every particle, we have a set of $N$ position vectors ${\bf r}_1,{\bf r}_2,...,{\bf r}_N$ and a set of $N$ momentum vectors ${\bf p}_1,{\bf p}_2,...,{\bf p}_N$. In order to follow the system in time, we clearly need to specify how each of the positions and momenta change as functions of time. This time evolution is given by Newton's second law of motion.

If ${\bf F}_1,{\bf F}_2,...,{\bf F}_N$ are the forces on particles 1, 2,...,N, respectively due to all of the other particles in the system, then the time dependence of all of the positions and momenta will be given by solving the set of equations of motion

$${\bf F}_i = m_i{d^2{\bf r}_i \over dt^2}\;\;\;\;\;\;\;\;\;\;i=1,...,N$$

In order to solve Newton's equations, all we have to do is specify all of the positions and momenta at one point in time, say at $t=0$, meaning that we have to provide the following vectors ${\bf r}_1(0),...,{\bf r}_N(0)$ and ${\bf p}_1(0),...,{\bf p}_N(0)$. This is called specifying the initial conditions. Once we have the initial conditions, then by solving the equations of motion, we know the time evolution of the system for all time in the future, and we can trace the history of the system infinitely far in the past.

More important, if we know all of the positions and momenta at all points in time, then the outcome of any experiment performed on the system can be predicted with perfect precision. Moreover, if the system is initially prepared in exactly the same way, with the same initial conditions in different repetitions of the experiment, then the same outcome will result every time.

Finally, since the positions and momenta of a system are continuous and can take on any values, physical quantities can also have any value since these are always computed from the positions and momenta. In particular, the energy

$$E = \sum_{i=1}^N {p_i^2 \over 2m_i} + V({\bf r}_1,...,{\bf r}_N)$$

can take on any value, although once its value is set for a given set of initial conditions, it will not change because energy is conserved.

Example: ``Classical Hydrogen''. Consider a hydrogen atom with the proton fixed at the origin and the electron a distance $r$ from the proton. The potential energy is

$$V(r) = -{e^2 \over 4\pi\epsilon_0 r}$$

which gives rise to a force

$${\bf F}= -{e^2 \over 4\pi \epsilon_0 r^3}{\bf r}$$

If we solve Newton's second law for the orbit of the electron, the orbit looks as shown in the figure below:

Figure 2:

Here, the energy $E$ of the orbit can take on any value.

Interestingly, according to Maxwell's theory of electromagnetism, an orbiting charged particle radiates energy due to its acceleration. Thus, the classical ``electron'' radiates energy as it orbits the proton. As a consequence, the electron loses energy continuously over time and eventually must spiral into the nucleus. Of course, this never happens, and this is one of the indications that classical mechanics has serious deficiencies.