Newton’s Laws in Translation

I’m about to do some lectures about Newton’s Laws of Motion to my first-year Mathematical Physics class so I thought I’d put up a quick post about how these laws have been expressed through the years. The original versions in the Principia (frontispiece above, first published in 1687) are of course in Latin. I did five years of Latin at school, but found most of the Principia impenetrable when I tried to read it in the original

 

The laws of motion are however fairly clear, perhaps because they are familiar in English:

Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus illud a viribus impressis cogitur statum suum mutare.

Lex II: Mutationem motus proportionalem esse vi motrici impressæ, & fieri secundum lineam rectam qua vis illa imprimitur.

Lex III: Actioni contrariam semper & æqualem esse reactionem: sive corporum duorum actiones in se mutuo semper esse æqualeset in partes contrarias dirigi.

As I am teaching in a room in the old college here in Maynooth (which was founded in 1795), I looked for a contemporary English translation. This is from 1792:

Law I: Every body perseveres in a state of being at rest or of moving uniformly straight forward except insofar as it is compelled to change its state by forces impressed.

Law II: The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

Law III: To every action there is always opposed an equal reaction: or the mutual action of two bodies upon each other are always equal, and directed to contrary parts.

And finally here’s the modern version I was taught at School:

First Law: Every body continues in a state of rest or uniform motion in a straight line unless it is acted upon by an external (unbalanced) force.

Second Law: The rate of change of momentum of a body is proportional to the impressed force, and is in the direction in which this force acts.

Third Law: To every action there is always an equal and opposite reaction,

an alternative form of the Third Law being:

Third Law: If Body A exerts a force on Body B then Body B exerts a force on Body A which is equal in magnitude and opposite in direction.

Going back to the 1792 English translation, the exposition of the second law continues:

If a force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

If only Newton had known vector notation!

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Why? You endeavoured to embroil me with weomen…

Here’s a post about an episode in the life of Sir Isaac Newton which I first came across when reading about Samuel Pepys. Many assume that Newton’s behaviour was a result of mental illness on his part, but that’s by no means clear. I can think of many possible reasons why he might have acted the way he did, including that he just found the behaviour of other people too perplexing…

Corpus Newtonicum

Why. It is a word that I frequently entertain when I study Isaac Newton. There is no scientist about whom so much is written, yet I feel that we only know so little about the man. Most Newton biographers provide us with detailed descriptions of his life and works, using the abundance of source materials available: Newton’s correspondence, descriptions by himself and others of various episodes of his life, Trinity College and Cambridge University attendance records, and so on. Every biographer, in his own way, tries to understand some of the more poignant moments in Newton’s life. Likewise, many struggle.

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The Cosmic Web

When I was writing my recent  (typically verbose) post about chaos  on a rainy saturday afternoon, I cut out a bit about astronomy because I thought it was too long even by my standards of prolixity. However, walking home this evening I realised I could actually use it in a new post inspired by a nice email I got after my Herschel lecture in Bath. More of that in a minute, but first the couple of paras I edited from the chaos item…

Astronomy provides a nice example that illustrates how easy it is to make things too complicated to solve. Suppose we have two massive bodies orbiting in otherwise empty space. They could be the Earth and Moon, for example, or a binary star system. Each of the bodies exerts a gravitational force on the other that causes it to move. Newton himself showed that the orbit followed by each of the bodies is an ellipse, and that both bodies orbit around their common centre of mass. The Earth is much more massive than the Moon, so the centre of mass of the Earth-Moon system is rather close to the centre of the Earth. Although the Moon appears to do all the moving, the Earth orbits too. If the two bodies have equal masses, they each orbit the mid-point of the line connecting them, like two dancers doing a waltz.

Now let us add one more body to the dance. It doesn’t seem like too drastic a complication to do this, but the result is a mathematical disaster. In fact there is no known mathematical solution for the gravitational three-body problem, apart from a few special cases where some simplifying symmetry helps us out. The same applies to the N-body problem for any N bigger than 2. We cannot solve the equations for systems of gravitating particles except by using numerical techniques and very big computers. We can do this very well these days, however, because computer power is cheap.

Computational cosmologists can “solve” the N-body problem for billions of particles, by starting with an input list of positions and velocities of all the particles. From this list the forces on each of them due to all the other particles can be calculated. Each particle is then moved a little according to Newton’s laws, thus advancing the system by one time-step. Then the forces are all calculated again and the system inches forward in time. At the end of the calculation, the solution obtained is simply a list of the positions and velocities of each of the particles. If you would like to know what would have happened with a slightly different set of initial conditions you need to run the entire calculation again. There is no elegant formula that can be applied for any input: each laborious calculation is specific to its initial conditions.

Now back to the Herschel lecture I gave, called The Cosmic Web, the name given to the frothy texture of the large-scale structure of the Universe revealed by galaxy surveys such as the 2dFGRS:

One of the points I tried to get across in the lecture was that we can explain the pattern – quite accurately – in the framework of the Big Bang cosmology by a process known as gravitational instability. Small initial irregularities in the density of the Universe tend to get amplified as time goes on. Regions just a bit denser than average tend to pull in material from their surroundings faster, getting denser and denser until they collapse in on themselves, thus forming bound objects.

This  Jeans instability  is the dominant mechanism behind star formation in molecular clouds, and it leads to the rapid collapse of blobby extended structures  to tightly bound clumps. On larger scales relevant to cosmological structure formation we have to take account of the fact that the universe is expanding. This means that gravity has to fight against the expansion in order to form structures, which slows it down. In the case of a static gas cloud the instability grows exponentially with time, whereas in an expanding background it is a slow power-law.

This actually helps us in cosmology because the process of structure formation is not so fast that it destroys all memory of the initial conditions, which is what happens when stars form. When we look at the large-scale structure of the galaxy distribution we are therefore seeing something which contains a memory of where it came from. I’ve blogged before about what started the whole thing off here.

Here’s a (very low-budget) animation of the formation of structure in the expanding universe as computed by an N-body code. The only subtlety in this is that it is in comoving coordinates, which expand with the universe: the box should really be getting bigger but is continually rescaled with the expansion to keep it the same size on the screen.

You can see that filaments form in profusion but these merge and disrupt in such a way that the characteristic size of the pattern evolves with time. This is called hierarchical clustering.

One of the questions I got by email after the talk was basically that if the same gravitational instability produced stars and large-scale structure, why wasn’t the whole universe just made of enormous star-like structures rather than all these strange filaments and things?

Part of the explanation is that the filaments are relatively transient things. The dominant picture is one in which the filaments and clusters
become incorporated in larger-scale structures but really dense concentrations, such as the spiral galaxies, which do
indeed look a bit like big solar systems, are relatively slow to form.

When a non-expanding cloud of gas collapses to form a star there is also some transient filamentary structure  but the processes involved go so rapidly that it is all swept away quickly. Out there in the expanding universe we can still see the cobwebs.

A Little Bit of Chaos

The era of modern physics could be said to have begun in 1687 with the publication by Sir Isaac Newton of his great Philosophiae Naturalis Principia Mathematica, (Principia for short). In this magnificent volume, Newton presented a mathematical theory of all known forms of motion and, for the first time, gave clear definitions of the concepts of force and momentum. Within this general framework he derived a new theory of Universal Gravitation and used it to explain the properties of planetary orbits previously discovered but unexplained by Johannes Kepler. The classical laws of motion and his famous “inverse square law” of gravity have been superseded by more complete theories when dealing with very high speeds or very strong gravity, but they nevertheless continue supply a very accurate description of our everyday physical world.

Newton’s laws have a rigidly deterministic structure. What I mean by this is that, given precise information about the state of a system at some time then one can use Newtonian mechanics to calculate the precise state of the system at any later time. The orbits of the planets, the positions of stars in the sky, and the occurrence of eclipses can all be predicted to very high accuracy using this theory.

At this point it is useful to mention that most physicists do not use Newton’s laws in the form presented in the Principia, but in a more elegant language named after Sir William Rowan Hamilton. The point about Newton’s laws of motion is that they are expressed mathematically as differential equations: they are expressed in terms of rates of changes of things. For instance, the force on a body gives the rate of change of the momentum of the body. Generally speaking, differential equations are very nasty things to solve which is a shame because most a great deal of theoretical physics involves them. Hamilton realised that it was possible to express Newton’s laws in a way that did not involve clumsy mathematics of this type. His formalism was equivalent, in the sense that one could obtain the basic differential equations from it, but easier to use in general situations. The key concept he introduced – now called the Hamiltonian – is a single mathematical function that depends on both the positions q and momenta p of the particles in a system, say H(q,p). This function is constructed from the different forms of energy (kinetic and potential) in the system, and how they depend on the p’s and q’s, but the details of how this works out don’t matter. Suffice to say that knowing the Hamiltonian for a system is tantamount to a full classical description of its behaviour.

Hamilton was a very interesting character. He was born in Dublin in 1805 and showed an astonishing early flair for languages, speaking 13 of them by the time he was 13. He graduated from Trinity College aged 22, at which point he was clearly a whiz-kid at mathematics as well as languages. He was immediately made professor of astronomy at Dublin and Astronomer Royal for Ireland. However, he turned out to be hopeless at the practicalities of observational work. Despite employing three of his sisters to help him in the observatory he never produced much of astronomical interest. Mathematics and alcohol seem to have been the two real loves of his life.

It is a fascinating historical fact that the development of probability theory during the late 17^th and early 18^th century coincided almost exactly with the rise of Newtonian Mechanics. It may seem strange in retrospect that there was no great philosophical conflict between these two great intellectual achievements since they have mutually incompatible views of prediction. Probability applies in unpredictable situations; Newtonian Mechanics says that everything is predictable. The resolution of this conundrum may owe a great deal to Laplace, who contributed greatly to both fields. Laplace, more than any other individual, was responsible to elevated the deterministic world-view of Newton to a scientific principle in its own right. To quote:

We ought then to regard the present state of the Universe as the effect of its preceding state and as the cause of its succeeding state.

According to Laplace’s view, knowledge of the initial conditions pertaining at the instant of creation would be sufficient in order to predict everything that subsequently happened. For him, a probabilistic treatment of phenomena did not conflict with classical theory, but was simply a convenient approach to be taken when the equations of motion were too difficult to be solved exactly. The required probabilities could be derived from the underlying theory, perhaps using some kind of symmetry argument.

The s-called “randomizing” devices used in all traditional gambling games – roulette wheels, dice, coins, bingo machines, and so on – are in fact well described by Newtonian mechanics. We call them “random” because the motions involved are just too complicated to make accurate prediction possible. Nevertheless it is clear that they are just straightforward mechanical devices which are essentially deterministic. On the other hand, we like to think the weather is predictable, at least in principle, but with much less evidence that it is so!

But it is not only systems with large numbers of interacting particles (like the Earth’s atmosphere) that pose problems for predictability. Some deceptively simple systems display extremely erratic behaviour. The theory of these systems is less than fifty years old or so, and it goes under the general title of nonlinear dynamics. One of the most important landmarks in this field was a study by two astronomers, Michel Hénon and Carl Heiles in 1964. They were interested in what would happens if you take a system with a known analytical solutions and modify it.

In the language of Hamiltonians, let us assume that H₀ describes a system whose evolution we know exactly and H₁ is some perturbation to it. The Hamiltonian of the modified system is thus

 

What Hénon and Heiles did was to study a system whose unmodified form is very familiar to physicists: the simple harmonic oscillator. This is a system which, when displaced from its equilibrium, experiences a restoring force proportional to the displacement. The Hamiltonian description for a single simple harmonic oscillator system involves a function that is quadratic in both p and q:

The solution of this system is well known: the general form is a sinusoidal motion and it is used in the description of all kinds of wave phenomena, swinging pendulums and so on.

The case Henon and Heiles looked at had two degrees of freedom, so that the Hamiltonian depends on q₁, q₂, p₁ and p_2:

 However, in this example, the two degrees of freedom are independent, meaning that there is uncoupled motion in the two directions. The amplitude of the oscillations is governed by the total energy of the system, which is a constant of the motion. Other than this, the type of behaviour displayed by this system is very rich, as exemplified by the various Lissajous figures shown in the diagram below. Note that all these figures are produced by the same type of dynamical system of equations: the different shapes are consequences of different initial conditions and different coefficients (which I set to unity in the form above).

 

 If the oscillations in each direction have the same frequency then one can get an orbit which is a line or an ellipse. If the frequencies differ then the orbits can be much more complicated, but still pretty. Note that in all these cases the orbit is just a line, i.e. a one-dimensional part of the two-dimensional space drawn on the paper.

More generally, one can think of this system as a point moving in a four-dimensional phase space defined by the coordinates q₁, q₂, p₁ and p₂; taking slices through this space reveals qualitatively similar types of orbit for, say, p₂ and q₂ as for p₁ and p₂. The motion of the system is confined to a lower-dimensional part of the phase space rather than filling up all the available phase space. In this particular case, because each degree of freedom moves in only one of its two available dimensions, the system as a whole moves in a two-dimensional part of the four-dimensional space.

This all applies to the original, unperturbed system. Hénon and Heiles took this simple model and modified by adding a term to the Hamiltonian that was cubic rather than quadratic and which coupled the two degrees of freedom together. For those of you interested in the details their Hamiltonian was of the form

 

 

The first set of terms in the brackets is the unmodified form, describing a simple harmonic oscillator; the other two terms are new. The result of this simple alteration is really quite surprising. They found that, for low energies, the system continued to behave like two uncoupled oscillators; the orbits were smooth and well-behaved. This is not surprising because the cubic modifications are smaller than the original quadratic terms if the amplitude is small.  For higher energies the motion becomes a bit more complicated, but the phase space behaviour is still characterized by continuous lines, as shown in the left hand part of the following figure.

 

However, at higher values of the energy (right), the cubic terms become more important, and something very striking happens. A two-dimensional slice through the phase space no longer shows the continuous curves that typify the original system, but a seemingly disorganized scattering of dots. It is not possible to discern any pattern in the phase space structure of this system: it appear to be random.

 

Nowadays we describe the transition from these two types of behaviour as being accompanied by the onset of chaos. It is important to note that this system is entirely deterministic, but it generates a phase space pattern that is quite different from what one would naively expect from the behaviour usually associated with classical Hamiltonian systems. To understand how this comes about it is perhaps helpful to think about predictability in classical systems. It is true that precise knowledge of the state of a system allows one to predict its state at some future time.  For a single particle this means that precise knowledge of its position and momentum, and knowledge of the relevant H, will allow one to calculate the position and momentum at all future times.

But think a moment about what this means. What do we mean by precise knowledge of the particle’s position? How precise? How many decimal places? If one has to give the position exactly then that could require an infinite amount of information. Clearly we never have that much information. Everything we know about the physical world has to be coarse-grained to some extent, even if it is only limited by measurement error. Strict determinism in the form advocated by Laplace is clearly a fantasy. Determinism is not the same as predictability.

In “simple” Hamiltonian systems what happens is that two neighbouring phase-space paths separate from each other in a very controlled way as the system evolves. In fact the separation between paths usually grows proportionally to time. The coarse-graining with which the input conditions are specified thus leads to a similar level of coarse-graining in the output state. Effectively the system is predictable, since the uncertainty in the output is not much larger than in the input.

In the chaotic system things are very different. What happens here is that the non-linear interactions represented in the Hamiltonian play havoc with the initial coarse-graining. Phase-space orbits that start out close to each other separate extremely violently (typically exponentially) and in a way that varies from one part of the phase space to another.  What happens then is that particle paths become hopelessly scrambled and the mapping between initial and final states becomes too complex to handle. What comes out  the end is practically impossible to predict.