CONSERVED QUANTITIES IN THE KEPLER–NEWTON PLANETARY MODEL

To an ancient man who watched the sky without nuances of smoke, dust and light contaminations in the atmosphere, the sky would have looked many times more beautiful and brighter than it does now. With amazing regularity, night after night, the sky would turn around his village. With no television to dissipate his time, he would admire and wonder, what is it that makes the sky look so very nearly the same each night, and yet that tad different. What a wonder it would seem that the world turned around him, every single day. A keen observer would notice, however, that amid the twinkling stars, there were some bright objects that seemed to be wandering a little bit in space, here today, and there tomorrow. Over days, weeks, and months, they would drift even far apart from the group of stars they were first sighted with. Astronomy is in some sense the mother of both physics and mathematics, ever since the curious man explored reasons to account for his observations. In studying astronomy, man hit on the very method of science, which would require geometry, trigonometry and eventually differential and integral calculus. Early models included imaginary forces, driven often by mythological gods and daemons, stories of whom fascinate children the world over even today. The myth, however, obliterates reason. Sections of the society sadly even now continue to be driven by superstition, rather than the knowledge earned by man over centuries. Today, much is known, and even as mindboggling questions continue to challenge physicists, superstition is thankfully becoming increasingly dispensable.

Star-gazing and analyzing motion of the planets, first considered as wandering stars, thus reveals the very method of scientific exploration. Human curiosity demands a model to be developed in order to account for the observed phenomena. One may trust the model as long as it does not lead to any discrepancy or contradiction, or internal inconsistency. A few early deductions known to some Indian and Greek astronomers have turned out to be quite accurate. For example, Aryabhatta, in the fifth century CE, had inferred and proclaimed that Earth has a spherical shape, not flat (as some people seem to want to believe even today).

THE VARIATIONAL PRINCIPLE AND EULER–LAGRANGE'S EQUATION OF MOTION

In the previous chapters, we have worked with the Newtonian formulation of classical mechanics. Its central theme relies on the use of ‘force’ as the very cause of change in momentum. The cornerstone of Newtonian mechanics is this principle of causality. It is expressed in Newton's second law as a linear relation between the acceleration and the force. It is the result of the equality between the force and the rate of change of momentum. The relation between force and momentum is at the very heart of Newtonian formulation of classical mechanics. It turns out that classical mechanics has an alternative but equivalent formulation, based on what is known as the ‘variational principle’, or ‘Hamilton's principle of variation’. In many universities, the principle of variation [1, 2] is introduced after a few years of college education in physics, and after a few courses on mechanics, including electrodynamics. However, there have been a few proposals [3, 4, 5] which recommend an early exposure in college curriculum to this fascinating approach. In fact, Richard Feynman was introduced to the principle of variation by his high school teacher, Mr Bader. Feynman went on to develop the path integral approach to the quantum theory based on the principle of variation. The path integral approach to quantum mechanics provides an alternative formulation of the quantum theory; it is equivalent to Heisenberg's uncertainty principle, and the Schrödinger equation. It has the capacity to describe a mechanical system and to account for how it evolves with time. The variational principle can be adapted to provide a backward integration of classical mechanics as an approximation toward the development of quantum theory. Newtonian formulation is not suitable for this purpose.

I am a theoretical astrophysicist, and my professional work requires a foundation in classical mechanics and fluid dynamics, and it then draws on statistical mechanics, relativity, and quantum mechanics. How is a student to see the underlying connections between these vast subjects? Professor Pranawa Deshmukh's book ‘Foundations of Classical Mechanics’ (FoCM) provides an excellent exposition to the underlying unity of physics, and is a valuable resource for students and professionals who specialize in any area of physics.

It was 2011 and I was Chair of the Department of Physics and Astronomy at the University of Western Ontario (UWO). An important global trend in education is internationalization: the effort to increase the international mobility of students and faculty, and increase partnerships in research and teaching. The present-day leading universities in the world are the ones that long ago figured out the benefits of academic mobility and exchanges. As part of our internationalization efforts at UWO, I was keen on building ties with the renowned Indian Institute of Technology (IIT) campuses in India. I invited Professor Deshmukh of the IIT Madras to come to UWO to spend a term in residence and also teach one course. That course turned out to be Classical Mechanics. How enlightening to find out that Professor Deshmukh had already developed extensive lecture materials in this area, including a full videographed lecture course ‘Special Topics in Classical Mechanics’ that is available on YouTube. Not surprising then, the course he taught at UWO was a tour-de-force of classical mechanics that also, notably, included topics that are considered to be ‘modern’, including chaos theory and relativity. Many of our students appreciated, highly, the inclusion of special relativity and, for example, to be able to finally understand a resolution to the mind-bending Twin Paradox. We were lucky to have Professor Deshmukh bring his diverse expertise to Canada, and UWO in particular, and it established personal and research links between individuals, and more generally between UWO and the IITs, which continue today.

The FoCM book is a further extension of the broad approach that Professor Deshmukh has brought to his teaching. This is not just another book on Classical Mechanics, due to both its approach and extensive content. The chapters are written in a conversational style, with everyday examples, historical anecdotes, short biographical sketches, and pedagogical features included.

THE SCALAR FIELD, DIRECTIONAL DERIVATIVE, AND GRADIENT

In the discussion on Fig. 2.4 (Chapter 2), we learned that it was neither innocuous to define a vector merely as a quantity that has both direction and magnitude, nor to define a scalar simply as a quantity that has magnitude alone. It is not that the properties referred here of a scalar and a vector are invalid. Rather, it is to be understood that these properties do not provide an unambiguous definition. Only a signature criterion of a physical quantity can be used to define it. We therefore introduced, in Chapter 2, comprehensive definitions of the scalar as a tensor of rank 0, and of the vector as a tensor of rank 1. In this chapter we shall acquaint ourselves with the mathematical framework in which the laws of fluid mechanics and electrodynamics are formulated using vector algebra and vector calculus. In fact, the techniques are used not merely in these two important branches of classical mechanics, but also in very various other subdivisions of physics. The background material seems at times to be intensely mathematical, but that is only because the laws of nature engage a mathematical formulation very intimately, as we encounter repeatedly in the analysis of physical phenomena. There are many excellent books in college libraries from which one can master the mathematical methods. These topics are extremely enjoyable to learn; they help us develop rigorous insights in the laws of nature. The literature on these topics is vast. A couple of illustrative books [1, 2] are suggested for further reading.

We consider the example of a particular scalar function, namely the temperature distribution in a room. The temperature is a physical property at a particular point in space, such as the point P in Fig. 10.1. Depending on the distribution of the sources of heat in the region that surrounds the point P, temperature may be different from point to point in space, and also possibly from time to time. The reason the temperature at a point is a scalar, is that its value at that point is independent of where the observer's frame of reference is located, and also independent of how it is oriented.

GEOMETRY OF THE SPACE–TIME CONTINUUM

The gravitational interaction is the earliest physical interaction that humans have registered. The earliest speculations about just what is the nature of gravity were not merely wrong, but absurdly far-fetched. Ancient philosophers even conjectured that the earth is the natural abode of things, and objects fall down when they are dropped just as horses return to their stables. Various theories of gravity were proposed, and the one that lasted much is that developed by Isaac Newton in the seventeenth century. Newton's work on gravity integrated the dynamics of astronomical objects with that of falling apples or coconuts, determined by one common principle (Chapter 8). We celebrate this principle as Newton's one-over-distance-square law of gravity.

An amazing consequence of the constancy of the speed of light in all inertial frames of reference that we studied in the previous chapter is the time-dilation and Lorentz contraction (also called the length contraction). The phenomenon that is responsible for the traveling twin to age slower than the home-bound twin holds for any and every object in motion. We have already noted that this happens to decaying muons. Essentially, the faster you move through space, the slower you move through time, in the spacetime continuum.

We all enjoy raising our speed, covering more distance in lesser, and lesser, time. Let us therefore ask, to what extent can we speed up an object? We ask if there is a natural limit for this. If you look back into the relations for time-dilation and the length contraction in the previous chapter, you will recognize that if v = c, the effect of time-dilation would be such that the traveling twin will simply stop ageing. Time would stop for her; time freezes. The effect of Lorentz contraction would also be total; she would think that the rest of the universe has spatially contracted to a point. She is therefore already everywhere (along the line of motion). All of these dramatic aftermaths are because of a simple fundamental property that the speed of the headlight of a car coming toward you at a velocity v is no different from that of the tail light of another that is receding away from you.

BOUNDED MOTION IN ONE-DIMENSION, SMALL OSCILLATIONS

Rise and fall, ups and downs, profit and loss, success and failure, happiness and sorrow—almost every experience in life seems to be in a state of oscillation. It is the undulations of the sound waves in air which enable us to hear each other, and it is undulations of the electromagnetic waves in vacuum which bring energy (light) from the stars to us. Quantum theory tells us that there is a wave associated with every particle, and that leaves us with absolutely nothing in the physical world that is not associated with oscillations. Oscillations of what, where and how can only be a matter of details, but there is little doubt that the physical universe requires for its rigorous study an understanding of ‘oscillations’. Oscillations are repetitive physical phenomena, which swing past some event, back and forth. They are ubiquitous, and they maintain things close to some mean behavior, at least most of the times. Whether it is the breeze jiggling the leaves, the rhythmic beating of the heart that is the acclamation of life itself, the gentle ripples on a river bed, or the mighty tides in the oceans, or for that matter fluctuations in the share market, we are always dealing with some expression of the simple harmonic oscillator, or a superposition of several of these, however, complex.

We have learned in earlier chapters that equilibrium, defined as motion at constant momentum, sustains itself when the object is left alone, completely determined by initial conditions. Departure from equilibrium is accounted for by Newton's equation of motion (second law), or equivalently (as we shall see in Chapter 6) by Lagrange's, or Hamilton's equations. In a 1-dimensional region of space, say along the X-axis, we consider a region in which the equilibrium of a particle may change because of spatial dependence of its interaction with the environment. This is represented by a space-dependent potential V(x).

THE EULER'S EQUATION OF MOTION FOR FLUID FLOW

In the previous chapter, we studied the Kelvin–Stokes theorem. When restricted to just two dimensions, it is referred to as the Green's theorem which is therefore only a special case of the Kelvin–Stokes theorem. Using the Green's theorem, we are automatically led to another prodigious theorem in the analysis of complex functions, known as the Cauchy's theorem. The methods developed in Chapter 10 are of great importance in fluid dynamics, electrodynamics and also in quantum dynamics. Practical applications of these methods are abundantly found not merely in fluid dynamics but also in the study of plasma in stellar atmosphere, and the analysis of charge plasma produced by lasers. The techniques are therefore of great consequence in engineering and technology, apart of course, from basic sciences. In this chapter, we shall use these methods to develop the equation of motion for a classical fluid.

The central question in mechanics is how to describe the state of a system, and how the system evolves with time. We have learned in earlier chapters that the state of a material particle system is represented by its position and momentum in the phase space. The temporal evolution of the system is provided by its equation of motion, viz., the Newton's equation of motion, or equivalently by the Lagrange or the Hamilton's equation of motion, which are discussed in earlier chapters. Much of the study of classical mechanics is about setting up the equation of motion, and learning to solve the same. When the medium is the continuum fluid, the system is very complex. It cannot be described as particles, and its evolution cannot be simply described by the usual familiar form of the Newton's, Lagrange's, or Hamilton's equations, even if the basic tenets of classical mechanics remain applicable. A fluid consists of a large number of fluid ‘particles’, which is an idea that is not defined exactly the same way as that for a piece of sand.