Pub Date : 2019-11-01DOI: 10.1017/9781108635639.013
P. Deshmukh
There is no greater burden than an unfulfilled potential. —Charles M. Schulz 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.
{"title":"Gradient Operator, Methods of Fluid Mechanics, and Electrodynamics","authors":"P. Deshmukh","doi":"10.1017/9781108635639.013","DOIUrl":"https://doi.org/10.1017/9781108635639.013","url":null,"abstract":"There is no greater burden than an unfulfilled potential. —Charles M. Schulz 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.","PeriodicalId":197751,"journal":{"name":"Foundations of Classical Mechanics","volume":"132 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114004644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-11-01DOI: 10.1017/9781108635639.011
P. Deshmukh
Because there is a law such as gravity, the universe can and will create itself from nothing. —Stephen Hawking 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).
{"title":"The Gravitational Interaction in Newtonian Mechanics","authors":"P. Deshmukh","doi":"10.1017/9781108635639.011","DOIUrl":"https://doi.org/10.1017/9781108635639.011","url":null,"abstract":"Because there is a law such as gravity, the universe can and will create itself from nothing. —Stephen Hawking 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).","PeriodicalId":197751,"journal":{"name":"Foundations of Classical Mechanics","volume":"72 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131849065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-11-01DOI: 10.1017/9781108635639.004
P. Deshmukh
{"title":"Laws of Mechanics and Symmetry Principles","authors":"P. Deshmukh","doi":"10.1017/9781108635639.004","DOIUrl":"https://doi.org/10.1017/9781108635639.004","url":null,"abstract":"","PeriodicalId":197751,"journal":{"name":"Foundations of Classical Mechanics","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123804844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-11-01DOI: 10.1017/9781108635639.015
P. Deshmukh
{"title":"Basic Principles of Electrodynamics","authors":"P. Deshmukh","doi":"10.1017/9781108635639.015","DOIUrl":"https://doi.org/10.1017/9781108635639.015","url":null,"abstract":"","PeriodicalId":197751,"journal":{"name":"Foundations of Classical Mechanics","volume":"510 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131409321","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-11-01DOI: 10.1017/9781108635639.017
P. Deshmukh
If at first the idea is not absurd, then there is no hope for it. —Albert Einstein 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.
{"title":"A Glimpse of the General Theory of Relativity","authors":"P. Deshmukh","doi":"10.1017/9781108635639.017","DOIUrl":"https://doi.org/10.1017/9781108635639.017","url":null,"abstract":"If at first the idea is not absurd, then there is no hope for it. —Albert Einstein 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.","PeriodicalId":197751,"journal":{"name":"Foundations of Classical Mechanics","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133523580","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-11-01DOI: 10.1017/9781108635639.012
P. Deshmukh
I am convinced that chaos research will bring about a revolution in natural sciences similar to that produced by quantum mechanics. — Gerd Binnig LEARNING FROM NUMBERS The equations of motion of classical mechanics, whether Newton's, Lagrange's, or Hamilton's, have us believe that given the state of the system at a particular time, one can always predict what its state would be any time later, or, for that matter, what it was any time earlier. This is because of the fact that the equations of motion are symmetric with respect to time-reversal: ( t )→(– t ). Classical mechanics relies on the assumption that position q and momentum p of a mechanical system are simultaneously knowable. Together, the pair ( q , p ) provides a signature of the state of the system . Their time-dependence, i.e., provided by the equations of motion, accurately describes their temporal evolution. For macroscopic objects, this is an excellent approximation, and the classical laws of mechanics are stringently deterministic. This is stringently correct, but there is an important caveat, expressed succinctly by Stephen Hawking: “ Our ability to predict the future is severely limited by the complexity of the equations, and the fact that they often have a property called chaos… a tiny disturbance in one place, can cause a major change in another. ” The difficulty Hawking alludes to has nothing to do with the quantum principle of uncertainty, but to a challenge within the framework of the fully deterministic classical theory. The solution to the temporal evolution may become chaotic , even as they remain deterministic, due to extreme sensitivity to the initial conditions that are necessary to obtain the solution to the equation of motion. Careful admission of the previous remark would prepare you to embark your journey on the exciting field of chaos. Along the way, you will also meet objects having weird fractional dimensions. The field covered by chaos theory is vast, though relatively young. It is a very rich field and can be introduced from a variety of perspectives. The general field of chaos theory is often regarded as a study of a ‘dynamical system’ which is just about any quantity which changes, and one has reasons to track these changes and the sequence of values it may take, for example, over a time interval.
{"title":"Complex Behavior of Simple System","authors":"P. Deshmukh","doi":"10.1017/9781108635639.012","DOIUrl":"https://doi.org/10.1017/9781108635639.012","url":null,"abstract":"I am convinced that chaos research will bring about a revolution in natural sciences similar to that produced by quantum mechanics. — Gerd Binnig LEARNING FROM NUMBERS The equations of motion of classical mechanics, whether Newton's, Lagrange's, or Hamilton's, have us believe that given the state of the system at a particular time, one can always predict what its state would be any time later, or, for that matter, what it was any time earlier. This is because of the fact that the equations of motion are symmetric with respect to time-reversal: ( t )→(– t ). Classical mechanics relies on the assumption that position q and momentum p of a mechanical system are simultaneously knowable. Together, the pair ( q , p ) provides a signature of the state of the system . Their time-dependence, i.e., provided by the equations of motion, accurately describes their temporal evolution. For macroscopic objects, this is an excellent approximation, and the classical laws of mechanics are stringently deterministic. This is stringently correct, but there is an important caveat, expressed succinctly by Stephen Hawking: “ Our ability to predict the future is severely limited by the complexity of the equations, and the fact that they often have a property called chaos… a tiny disturbance in one place, can cause a major change in another. ” The difficulty Hawking alludes to has nothing to do with the quantum principle of uncertainty, but to a challenge within the framework of the fully deterministic classical theory. The solution to the temporal evolution may become chaotic , even as they remain deterministic, due to extreme sensitivity to the initial conditions that are necessary to obtain the solution to the equation of motion. Careful admission of the previous remark would prepare you to embark your journey on the exciting field of chaos. Along the way, you will also meet objects having weird fractional dimensions. The field covered by chaos theory is vast, though relatively young. It is a very rich field and can be introduced from a variety of perspectives. The general field of chaos theory is often regarded as a study of a ‘dynamical system’ which is just about any quantity which changes, and one has reasons to track these changes and the sequence of values it may take, for example, over a time interval.","PeriodicalId":197751,"journal":{"name":"Foundations of Classical Mechanics","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123795494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-11-01DOI: 10.1017/9781108635639.008
P. Deshmukh
Lost time is never found again. —Benjamin Franklin DISSIPATIVE SYSTEMS Energy dissipative systems prompt us to think of physical phenomena in which energy is not conserved, it is lost. We attribute these losses to ‘friction’, which is the common term used to describe energy dissipation. Now, in our everyday experience, we are primarily involved with the gravitational and electromagnetic interactions, and both of these are essentially conservative. What is it, then, that makes friction non-conservative ? What does non-conservation of energy, or ‘energy-loss’, really mean? Fundamental interactions in nature allow energy to be changed from one form to another, but not created or destroyed. It therefore seems that the term ‘energy loss’ is used somewhat loosely. We must be really careful when we talk about dissipative phenomena. Losing money is often a matter of concern, as also other things we sometimes ‘lose’ from time to time, including time itself. Time wasted does not ever come back, of course; but nor does the time well-spent. The difference between the two is that the latter is accounted for by the gains made, and for the former there is simply no account. Isn't it merely a matter of book-keeping? It is not at all uncommon that we plan to do something during the day, and end up not doing it. We then sit and wonder where lost track of time. Did the day just skip over the afternoon and ring in the evening? If that did not happen, where was the time ‘lost’? You possibly remember everything that you did since morning, and you may be able to account for every hour you spent, except perhaps for what happened between 2:30 pm and 3 pm. That was the time you ransacked your house to find your mathematics text book, and did not realize that it took you half an hour to find it. The book had not really vanished, even if you suspected that it was lost. You had to look for it all over, from your Dad's room to your Sister's. As such, neither the book was lost, nor was half an hour scooped out of the day.
{"title":"Damped and Driven Oscillations; Resonances","authors":"P. Deshmukh","doi":"10.1017/9781108635639.008","DOIUrl":"https://doi.org/10.1017/9781108635639.008","url":null,"abstract":"Lost time is never found again. —Benjamin Franklin DISSIPATIVE SYSTEMS Energy dissipative systems prompt us to think of physical phenomena in which energy is not conserved, it is lost. We attribute these losses to ‘friction’, which is the common term used to describe energy dissipation. Now, in our everyday experience, we are primarily involved with the gravitational and electromagnetic interactions, and both of these are essentially conservative. What is it, then, that makes friction non-conservative ? What does non-conservation of energy, or ‘energy-loss’, really mean? Fundamental interactions in nature allow energy to be changed from one form to another, but not created or destroyed. It therefore seems that the term ‘energy loss’ is used somewhat loosely. We must be really careful when we talk about dissipative phenomena. Losing money is often a matter of concern, as also other things we sometimes ‘lose’ from time to time, including time itself. Time wasted does not ever come back, of course; but nor does the time well-spent. The difference between the two is that the latter is accounted for by the gains made, and for the former there is simply no account. Isn't it merely a matter of book-keeping? It is not at all uncommon that we plan to do something during the day, and end up not doing it. We then sit and wonder where lost track of time. Did the day just skip over the afternoon and ring in the evening? If that did not happen, where was the time ‘lost’? You possibly remember everything that you did since morning, and you may be able to account for every hour you spent, except perhaps for what happened between 2:30 pm and 3 pm. That was the time you ransacked your house to find your mathematics text book, and did not realize that it took you half an hour to find it. The book had not really vanished, even if you suspected that it was lost. You had to look for it all over, from your Dad's room to your Sister's. As such, neither the book was lost, nor was half an hour scooped out of the day.","PeriodicalId":197751,"journal":{"name":"Foundations of Classical Mechanics","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116811147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-11-01DOI: 10.1017/9781108635639.007
P. Deshmukh
I will be the waves and you will be a strange shore. I shall roll on and on and on, and break upon your lap with laughter. And no one in the world will know where we both are. —Rabindranath Tagore 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 ).
{"title":"Small Oscillations and Wave Motion","authors":"P. Deshmukh","doi":"10.1017/9781108635639.007","DOIUrl":"https://doi.org/10.1017/9781108635639.007","url":null,"abstract":"I will be the waves and you will be a strange shore. I shall roll on and on and on, and break upon your lap with laughter. And no one in the world will know where we both are. —Rabindranath Tagore 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 ).","PeriodicalId":197751,"journal":{"name":"Foundations of Classical Mechanics","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114713906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-10-31DOI: 10.1017/9781108635639.006
{"title":"Real Effects of Pseudo-forces: Description of Motion in Accelerated Frame of Reference","authors":"","doi":"10.1017/9781108635639.006","DOIUrl":"https://doi.org/10.1017/9781108635639.006","url":null,"abstract":"","PeriodicalId":197751,"journal":{"name":"Foundations of Classical Mechanics","volume":"102 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134173731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-10-31DOI: 10.1017/9781108635639.009
P. Deshmukh
When I was in high school, my Physics teacher—whose name was Mr Bader—called me down one day after physics class and said, ‘You look bored; I want to tell you something interesting’. Then he told me something which I found absolutely fascinating, and have since then, always found fascinating. Every time the subject comes up, I work on it… The subject is this—the principle of least action. —Richard P. Feynman 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 Schrodinger 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.
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