Throughout the different entries published in this section we have been showing how mathematics is the necessary language to understand the universe, as Galileo Galilei explicitly wrote in Il Saggiatore in 1623. This idea was exploited to the full by Isaac Newton with his Principia Mathematica of 1687 who, as we have already mentioned, laid the foundations of a method that we continue to use today with enormous success: what has come to be called Newton’s programme.
The success of Newton’s programme of explaining natural phenomena using mathematical language has been resounding. After Newton, countless scientists (many of whom were more than physicists or mathematicians) began to use the tools he had made available to them: differential and integral calculus, or as it is also known: infinitesimal calculus. The infinitesimal calculus discovered (or developed, as you prefer) by Newton and Gottfried Leibniz (who also discovered it independently) is the tool that allows us to deal with infinitely small (but non-zero) numbers and is essential for understanding the motion of bodies. Thanks to Newton’s laws we can establish the dynamical equations describing the motion of a particle. And why do we need this new calculus? To understand it, let us consider a very simple problem: the motion of a particle of mass \(m\) under the action of a force \(F\) that depends on the position of the particle (two very simple systems are the simple harmonic oscillator or the motion of a planet around the sun). For simplicity let us imagine that the particle is moving in a straight line. The force acting on the particle is a function of the position \(x\). The dynamic equation describing the motion of the first particle is given by Newton’s second law
$$ F(x(t))= m a(t) $$
The above tells us that the acceleration \(a\) of our particle is going to change with the distance, that is to say, in each moment of time both the distance \(x\) and the acceleration \(a\) change. Now, as we know, the acceleration and the position of our particle are not independent quantities, there is a very important relationship between them. The acceleration is the derivative of the velocity of the particle and therefore the second derivative of the position. Thus, if \(x(t)\) represents the position of our particle at time instant \(t\), then the velocity is given by the derivative of the function \(x(t)\) defined by the expression (we owe the notation \(\displaystyle \frac{dx(t)}{dt}\) to Leibniz).
$$ v(t)=\displaystyle \frac{dx(t)}{dt} = \lim_{\Delta t\to0}\frac{x(t+\Delta t)-x(t)}{\Delta t}\quad \mbox{y}\quad a(t)\displaystyle=\frac{d^2x(t)}{dt^2} =\displaystyle\lim_{\Delta t\to0}\frac{v(t+\Delta t)-v(t)}{\Delta t}$$
In this way, what we would obtain is a differential equation, that is, an equation whose unknown is a function (and not a number).
$$ m\frac{d^2x(t)}{dt^2} = F(x(t))$$
For example, in the case of the harmonic oscillator the previous equation has the form \(\frac{d^2x}{dt^2} + \frac{k}{m} x=0\), whose solution will depend on the initial conditions \(x(t_0)=x_0\) and \(v(t_0)=v_0\) at the time instant \(t_0\).
In general, if we have a certain number \(N\) of particles in space, they will be governed by a system of \(3N\) differentiable equations with the form
$$ m_k\frac{d^2 x_k}{dt^2}+F_{x,k}=0,\quad m_k\frac{d^2 y_k}{dt^2}+F_{y,k}=0, \quad m_k\frac{d^2 z_k}{dt^2}+F_{z,k}=0, $$
where \(x_k,y_k,z_k\) are the coordinates of the positions of each of the particles, \(k=1,2,\dots,N\) and \(F_{x,k}, F_{y,k}, F_{z,k}\), the corresponding forces on each of the axes. This is what is known in mathematics as a system of ordinary differential equations. Under certain reasonable assumptions it can be proved that, given the initial conditions for our system, i.e. the values \(x_k(0), y_k(0), z_k(0)\) and those of their derivatives \(\frac{d}{dt}x_k(0), \frac{d}{dt}y_k(0), \frac{d}{dt}z_k(0)\), the solution is unique. That is, if we know the state of a system at a given time (\(t=0\) in our example) we can know what happens to it at any time before or after. In the words of Max Born in his Nobel Prize acceptance lecture (December 1954)
Newtonian mechanics is deterministic in the following sense: If the initial state (positions and velocities of all particles) of a system is accurately given, then the state at any other time (earlier or later) can be calculated from the laws of mechanics. All the other branches of classical physics have been built up according to this model. Mechanical determinism gradually became a kind of article of faith: the world as a machine, an automaton. As far as I can see, this idea has no forerunners in ancient and medieval philosophy. The idea is a product of the immense success of Newtonian mechanics, particularly in astronomy. In the 19th century it became a basic philosophical principle for the whole of exact science.
This idea was not new in physics. Several physicists/mathematicians had already asked the same question. Pierre-Simon de Laplace (1749-1827), one of the best mathematicians of the late 17th and early 19th centuries, demonstrated the stability of the solar system by applying Newton’s laws. This problem had plagued physicists ever since Newton, since the motion of the planets, especially those of Jupiter and Saturn, appeared to deviate from what was to be expected according to Newtonian theory. To force everything to be in order Newton himself wrote (see, for example, the end of the segunda edición de su Óptica de 1717, page 378, in the 1952 edition of the Encyclopedia Britannica)
For while Comets move in very excentrick Orbs in all manner of Positions, blind Fate could never make all the Planets move one and the fame way in Orbs concentrick, some inconsiderable Irregularities excepted, which may have risen from the mutual Actions of Comets and Planets upon one another, and which will be apt to increase, till this System wants a Reformation. [which in the context and the way it is written the reformation or correction will have to come from the hand of God].
Laplace proved between 1770-1790 that these irregularities were periodic and therefore after a certain time the positions of the planets would be repeated, i.e. the solar system was stable. In other words, Laplace was convinced that Newton’s equations could perfectly explain biunivocally any phenomenon that we can observe in nature. This led him to write in his famous essay “Essai philosophique sur les probabilités” (see pages 3 and 4 of the 1812 edition) the following:
An intelligence which, at a given instant, knew all the forces which animate nature, and the respective situation of the beings of which it is composed, if it were elsewhere sufficiently complex to submit these data to analysis, would include in the same formula the movements of the largest bodies in the universe and those of the lightest atom: nothing would be uncertain to it, and the future, like the past, would be present to its eyes.
Such intelligence has passed to posterity under the name of Laplace’s demon. There is abundant literature on the possibility or not that a Laplace’s demon could exist in our Universe, but that would lead us away from the purpose of this entry. What is not in doubt, at least with current data and calculations, is that our solar system is sufficiently stable. A superb exposition of the subject can be found in [1].
The underlying problem we have to take into account is the assumption that we can know all the positions and velocities of each and every one of the particles that make up the Universe, something difficult to conceive of as anyone can imagine, even if we restrict ourselves to classical physics (we already know that the atomic world is governed by very different laws, the laws of quantum mechanics where Heisenberg’s uncertainty principle plays a fundamental role).
So, like Laplace, let us assume that it is possible to know the state of the Universe, i.e. a priori we can measure all the positions and velocities of all the particles in the Universe, or of any of the physical quantities we want to measure. To be able to compare our theoretical results with the observational ones, all we need is more and more sophisticated apparatus. In fact, at the end of the 19th century, physicists were convinced that the future of physics lay in being able to measure with greater precision. Thus, for example, James Clerk Maxwell (already mentioned here) in his introductory lecture on Experimental Physics in 1871 stated [2]:
This feature of modern experiments, which consist mainly of measurements, is so prominent that the opinion seems to have spread that in a few years all the great physical constants will have been approximately estimated, and that the only occupation left to men of science afterwards will be to continue these measurements by increasing their precision to another place of decimals.
This view would later be reinforced by the words of the eminent American experimental physicist A. A. Michelson, who in 1894 stated (Annual Register of the University of Chicago, 1896, p. 159):
Whereas it is never safe to say that the future of physical science will not hold wonders even more astonishing than those of the past, it seems likely that most of the great underlying principles have been firmly established, and that further advances will be sought mainly in the rigorous application of these principles to all the phenomena before us. It is here that the science of measurement shows its importance, where quantitative work is more desirable than qualitative work. An eminent physicist remarked that the future truths of physical science must be sought in the sixth decimal place.
Such an eminent physicist may well have been Lord Kelvin, whom we have already discussed here y here.
However, a very simple reasoning allows us to discover that the situation is much more complicated than Laplace supposed. So continued Born in his Nobel Price acceptance lecture which we mentioned above:
I asked myself whether this was really justified. Is it really possible to make absolute predictions for all times on the basis of the classical equations of motion? It can easily be seen, by simple examples, that this is only the case when the possibility of a completely exact measurement (of position, velocity or other quantities) is given. Think of a particle moving frictionlessly in a straight line between two walls, where it experiences a completely elastic collision. It moves with constant velocity equal to its initial velocity \(v_0\) forwards and backwards, and it can be known exactly where it will be at any time \(t\) if we know precisely the velocity \(v_0\). But if one were allowed a small inaccuracy in the velocity \(\Delta v_0\), then the inaccuracy in predicting the position at any instant would be \(\Delta v_0 t\) and would increase with time \(t\). If one waits long enough, \(t_c=l/\Delta v_0\), then the inaccuracy \(\Delta x\) will have become the total distance \(l\) between the walls. It is therefore impossible to predict anything about the position in a sufficiently long time greater than \(t_c\). Therefore, determinism becomes indeterminism from the moment the smallest imprecision in the velocity data is allowed.
In other words, even in the simplest case of a body moving frictionlessly between two walls, the formal impossibility of measuring its velocity with infinite precision leads to the impossibility of knowing where it will be after a very long time. It is clear that once again we have come up against a philosophical question for which physical arguments alone are not decisive, as Born himself wrote in 1926 (and which we already quoted in a previous post).
But let us assume for a moment that we are able to measure with absolute precision. Could we then, like Laplace, be sure that we will know everything at all times? Do Newton’s equations really describe a deterministic world?
Unfortunately, the answer is no. The one who proved it was another great French mathematician Jules Henri Poincaré (1852-1912), considered by many to be the last universal mathematician, a genius if ever there was one, with impressive contributions in practically all areas of mathematics, from analysis, geometry, topology, algebra, etc.
The story in a nutshell is as follows: On the occasion of the 60th birthday of King Oscar II of Sweden a mathematical competition was opened as one of the many commemorations of the event. A problem was supposed to be solved out of four problems mostly suggested by Karl Weierstrass. One of them had to do with the stability of the solar system, which we have already discussed. Specifically, the problem was the following:
Given an arbitrary system of many points of mass attracting each other according to Newton’s law, under the assumption that two points never collide, try to find a representation of the coordinates of each point as a series in one variable which is a known function of time and for all whose values the series converges uniformly.
In mathematical terms, Laplace’s proof did not pass the rigorous test imposed by Weierstrass. The fundamental reason was that the solutions of Newton’s equations could not be solved exactly and the mathematicians/physicists found approximations by infinite series of which they kept the first terms. Examples of such series include, for example, the following
$$ 1+x+\frac{x^2}{2}+\cdots +\frac{x^n}{n!}+\cdots ,\quad 1+x+2x^2+6x^3+\cdots+n! x^n +\cdots. $$
The first of the series converges for all real \(x\), it is a convergent series, while the second diverges for all \(x\neq0\), that is, it is a divergent series. The mathematical problem of stability was then reduced, roughly speaking, to whether the solution was expressed as a convergent or divergent series. The reason is that for the calculations it was impossible to consider all the terms of the series (which are infinite), so only the first ones were taken and the rest were disregarded. This procedure is dangerous if we do not know the nature of the series, because although for convergent series it is known that adding consecutive terms changes the final result very little, this is not the case for divergent series, where the addition of several terms leads to increasingly larger sums.
One of the papers received by the committee in charge of deciding who would win the generous prize was that of the still young French mathematician Henri Poincaré, who was 35 years old but who already enjoyed some recognition in his native France (he had been a professor at the University of Paris since the age of 27). The paper was entitled “Sur le problème des trois corps et les équations de la dynamique” (On the Three-body problem and the equations of dynamics) and in it Poincaré developed a whole new geometrical theory that gave him a global idea of the problem and answered the question of stability, although in his original article submitted in May 1888, for which he was awarded the prize, an error was detected, which he himself later corrected after two years of intense work and finally saw the light of day in 1890, in the journal Acta Mathematica. Weierstrass himself acknowledged the French mathematician’s work in this way: “This work cannot be considered to provide the complete solution of the proposed question, but it is of such importance that its publication will inaugurate a new era in the history of celestial mechanics”.
Poincaré’s paper was 270 pages long and had an influence on physics and mathematics comparable to Joseph Fourier’s famous 1822 treatise on heat for its originality and all the new areas that came to light after its publication. As far as we are concerned in relation to our post, Poincaré’s final conclusion could not have had greater repercussions: he implanted in the heart of Newtonian mechanics the uncertainty that Heisenberg had already implanted in quantum mechanics. In other words, Laplace’s deterministic dream was impossible. What better way to end this entry than with Ivars Peterson’s description of Poincaré’s work in his book Newton’s Clock, Chaos in the Solar System [4].
Any reader who dared to challenge his difficult language, unconventional strategies and daunting complexities experienced a mind-boggling but highly rewarding ride through a new mathematics with startling implications for the mathematical modelling of physical phenomena. With devastating effect, Poincaré systematically demolished the structure so dear to Weierstrass. He introduced doubt and uncertainty where the old man (Weierstrass) had anticipated a neat mathematical solution that would pave the way to perpetual certainty.
And Peterson went on
Poincaré began by stating that, although the equations modelling the motion of three gravitationally interacting bodies produce a well-defined relationship between time and position, there is no general computational shortcut – no magic formula – for making accurate predictions of position in the distant future. In other words, the series that emerge from perturbation theory tend to diverge. In fact, there was plenty of room for unpredictability in a Newtonian system, and the question of stability could not be solved directly by examining the divergent series associated with the solutions of the equations of motion of the solar system.
Chaos theory had just been born (discussed a bit in this other post), and Poincaré was undoubtedly the father of the creature, a theory that is masterfully described by the question with which the meteorologist Edward Norton Lorenz titled his famous 1972 lecture: Can a butterfly flapping its wings in Brazil produce a tornado in Texas? But this is a story for another entry.
Bibliography
[1] Laskar J. (2013) Is the Solar System Stable?. In: Duplantier B., Nonnenmacher S., Rivasseau V. (eds) Chaos. Progress in Mathematical Physics, vol 66. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0697-8_7 (see https://arxiv.org/abs/1209.5996 )
[2] The Scientific Papers of James Clerk Maxwell, ed. by W.D. Niven, vol. II (New York: Dover, 1965), pp. 241-255.
To learn more about Laplace, see
[3] Ana Rioja y Javier Ordóñez, Teorías del universo. Vol III, Ed. Síntesis, 2006 (sección 4.4 pág 211 y siguientes).
The story of Poincaré’s article is masterfully narrated in
[4] Ivars Peterson, El reloj de Newton, Caos en el sistema solar, Alianza Editoarial, 1995 (capítulo 7).
[5] On the \(n\)-body problem there is an excellent post by Juan Arias in this very blog.
Final comments
Poincaré’s manuscript corrected and resubmitted in 1889 can be downloaded here.
The featured image is built from figure 7 of Poincaré’s manuscript, a portrait of Laplace and a photograph of Poincaré.
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