The predominant usage of Green function is in the possesive form (i.e., Green's function). As a physicist, I find myself conforming to this as well, however, I was corrected by my Ph.D. advisor who always insisted that that was incorrect. "It is Bessel function, not Bessel's function, and so it should be with Green." (I am paraphrasing her here.) Is this true? I notice that Jackson (of Classical Electrodynamics) refers consistently to Green function, as does Eyger and Cohen-Tanoudji. Others (Fetter and Walecka, Arfkin, Morse and Fechbach, etc.) consistently use Green's function. Many others, I have found, interchange them, as do you, e.g. Physics Today 2003 uses Green function throughout, while your review of Mary Cannell's biography of Green uses Green's function.
Where is the first citation where the term is coined? I've read Green's work and he doesn't coin the term himself. I presume it was Lord Kelvin, but I cannot find a reference for this. Is there a consistency in the symantics in that first publication?
While this is an inconsequential issue, it has been one that has irritated me for some time. I would be very happy to put this topic to rest once and for all in my eyes.
Also, what would be your recommendation of the best George Green biography for a physicist to read?
Thank you,
TJ Ulrich, Ph.D.
Los Alamos National Laboratory
Thursday, 7 June 2007
Friday, 1 June 2007
George Green – Miller, Mathematician and Physicist
by L.J. Challis, University of Nottingham
This article is reproduced by permission of the Applied Probability Trust .from Mathematical Spectrum, 20(1987/8), 45-52
Copyright (c) The Applied Probability Trust 1987.
Reproduced with permission.
http://www.appliedprobability.org/
To avoid confusion, the telephone number of Green’s mill has been updated.
Lawrie Challis was Professor of Low Temperature Physics at the University of Nottingham. His main interests were experimental studies of electrons in semiconductors using very high frequency sound at low temperatures. He was also chairman of the George Green Memorial Fund which contributed to the restoration of Green’s windmill, now open to the public and grinding corn again.
Introduction
George Green (1793-1841) was a pioneer in the application of mathematics to physical problems. He was a miller who lived in Nottingham virtually all his life and had very little formal education until he had completed most of his best work. Partly as a result of his unusual circumstances he received little public recognition in his lifetime, and it was William Thomson (Lord Kelvin) who first recognised the value of his work and gave it wide publicity. His work has had great influence and nowadays he is remembered principally for Green’s theorem in vector analysis, Green’s tensor (or the Cauchy-Green tensor) in elasticity theory and above all for Green’s functions for solving differential equations. The Green’s function technique has been very widely applied to equations arising in classical physics and engineering and in recent years has been adapted to quantum mechanical problems in areas as diverse as nuclear physics, quantum electrodynamics and superconductivity. Yet until quite recently Green was a shadowy figure whose name was practically unknown to most non-scientists even in his home town. All this has now changed and, with the help of many organisations and of interested people from many countries, the City of Nottingham has created a splendid and living memorial to George Green. They have restored his ruined windmill to full working order and built a Science Centre illustrating some of his many interests.
George Green’s life
George Green was born in Nottingham on 13 July 1793. For generations his ancestors had been farmers in the village of Saxondale just a few miles from Nottingham, but his father, the youngest of three sons, had been sent there in 1774 to be apprenticed to a baker in Nottingham. In time he bought his own bakery and prospered, acquiring both land and property, which he rented out, as well as a warehouse on the banks of the River Lean which he used to store grain before sending it to be milled into flour for his bakery. When George was eight, he was sent to Robert Goodacre’s Academy. His schooling lasted only four terms, however, and then he left to help in the bakery. He was lucky in that his father sent him to that particular school, as Robert Goodacre was an enthusiastic science teacher. Indeed, later in his life, Goodacre became a popular speaker on astronomy, lecturing throughout the British Isles and America. So George Green would have acquired a taste for science, although it is doubtful whether his teacher could have stretched him very far in mathematics; Goodacre had had no formal training and had been apprenticed to a tailor before becoming a schoolmaster.
So at the age of nine, George Green had received all the formal education he was going to get until he was 40! There were of course bookshops in Nottingham where he could buy textbooks and encyclopedias, but there were as yet no libraries. It is possible that he may have received some guidance in his reading from one of the graduate mathematicians living in Nottingham.
When Green was 14, his father built a windmill at Sneinton, then a separate village a mile or so from Nottingham. It was a fine five-storey brick tower mill with stables for eight horses and storage for hay and corn. Milling was a skilled trade and he employed a foreman-manager, William Smith, who lived in a cottage attached to the side of the mill. The mill could not easily be worked single-handed; George helped William Smith, and so learned to operate the mill himself. This must have been an exciting change from the bakery for a boy of 14, and it would have been a strenuous, mostly outdoor life. When there was wind enough, he would have worked long hours even at night trimming the sails, maintaining the inflow of grain and taking away the milled flour or cattle fodder. As the Greens were still living in Nottingham, it seems likely that George might have stayed overnight with the Smiths or perhaps slept in the mill rather than walking back through the dark and probably dangerous streets of overcrowded Nottingham. It seems likely too that during this time he would have spent some of the calm days studying mathematics while waiting for the wind to come. Certainly his youngest daughter, Clara, who lived until 1919, told Professor Granger of University College, Nottingham, that her father used the top floor of the mill as a study.
When George was 24, he and his parents moved to a five-bedroomed house they built next to the mill and a few years later he joined the recently-opened Nottingham Subscription Library. This soon became the centre of intellectual life in Nottingham. It contained a modest collection of mathematical and scientific textbooks, and, of great importance, it took the important British scientific journals. These usually also included the titles and abstracts of papers from foreign journals, so that Green would have been able to follow what was being done elsewhere. In principle, he could then have written to the authors asking for copies of their papers.
In 1828, at the age of 35, George Green published his first paper "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism". It was a major work of striking originality. He invented completely new mathematical techniques to solve the problems that arose in the analysis and it would have had an immediate and profound effect had it been read by others working in the field. Tragically, it did not have this effect until some years after his death. He was advised that as he had had no formal training and his social position was modest, he could not presume to send his paper to a scientific journal! So instead he had it printed privately in Nottingham and presumably distributed a few copies to other mathematicians and physicists working in Britain. It had virtually no impact; hardly anyone in Britain had worked in this field for 50 years. British mathematicians were interested in mechanics, optics, astronomy, planetary motion and hydrodynamics; Green’s inspiration came from France, from Laplace and Poisson, but nobody there appears to have seen his paper. This lack of response must have depressed Green, but he soon started work on a second paper. He received valuable encouragement from Sir Edward Bromhead, an influential Cambridge mathematics graduate who lived in Lincolnshire and clearly realised Green’s exceptional ability.
Green turned to areas of much more interest to British mathematical physicists and, with Bromhead’s connections as his passport, he started to publish papers in the scientific journals. His family life also changed considerably about this time, when his father died. His mother had died some years before, so Green became a rather wealthy man. He stopped milling, leasing out his mill, and then in 1833, at the age of 40, he realised what may have been his ambition for many years. With Bromhead’s help he entered Cambridge as an undergraduate to read for a degree in mathematics. He took his degree in 1837 and shortly afterwards was elected to a fellowship at his college, Gonville and Caius.
Sadly, he held the post for only two years before becoming ill and returning to Nottingham, where he died in 1841. Sadly too, the full value of his work was not appreciated until after his death. Indeed, the only obituary that appeared was in a local Nottingham paper. It was three sentences long and ends "Had his life been prolonged, he might have stood eminently high as a mathematician".
Green’s mathematics
Green’s mathematics was nearly all devised to solve very general physical problems. His first interest was in electrostatics. The inverse-square law had recently been established experimentally, and he wanted to calculate how this determined the distribution of charge on the surfaces of conductors. He made great use of the electrical potential, and gave it that name, and one of the theorems that he proved in this work became famous and is known as Green’s theorem. It relates the properties of mathematical functions at the surfaces of a closed volume to other properties inside. In its usual form, the theorem involves two functions, but it readily simplifies to what is often called the divergence theorem or Gauss’s theorem. (Many early textbooks also called this form Green’s theorem as well, presumably to emphasise his claims to precedence).
To illustrate the theorem, we consider gas leaking from holes in the walls of a gas cylinder. The mass leaving per second per unit area equals the product of the density of the gas and its velocity at each hole. So we can find the total loss rate by integrating this over all the holes. (The integral can in fact be carried out over the whole surface since the contribution from the rest is zero). But this loss rate from the surface must equal the sum of the masses leaving per second from all the small volume elements dV inside the surface and this can be found by integrating a particular function over the whole volume V. The function is the result of a differential operator 4 called the divergence acting on the product of density and velocity of the gas at the element dV. The theorem relating the integral over the surface to the integral over the volume inside is useful in many branches of physics. For example, in electrostatics, a development closely related to this links the electric flux leaving a surface to the total charge inside it.
Another powerful technique invented by Green is used for solving differential equations. It involves what are now called Green’s functions, G(x,x’). If we have a differential equation Ly = F(x), where L is a linear differential operator, then the solution can be written
To see this, consider the equation
This can be solved by the standard integrating factor technique to give
so that G(x,x’) = exp[-k(x-x')].
To understand the meaning of this in a physical situation, consider the motion of a unit mass (initially at rest) following the application of a time-varying force F(t). If the motion is damped by a force –kv then, from Newton’s law,
with solution
as before. This can be interpreted by visualising the time-varying force F(t) as a rapid sequence of sharp blows each acting for a short time dt’ and causing a change in momentum or impulse F(t’)dt’. Thus the velocity v(t) at time t is the sum (or integral) of the effects of all blows from t’ = 0 to t’ = t. The velocity at time t due to a single impulse (with unit change of momentum) applied at time t’ is called Green’s function
This technique may be applied to other more complicated systems. In an electrical circuit the Green’s function is the current due to an applied voltage pulse. In electrostatics the Green’s function is the potential due to a change applied at a particular point in space. In general the Green’s function is the response of a system to a stimulus applied at a particular point in space or time. This concept has been readily adapted to quantum physics where the applied stimulus is the injection of a quantum of energy. It is in the quantum domain that the application of Green’s functions to physical problems has grown most spectacularly in the past few decades.
Green also did very original work on elasticity, where he is remembered by Green’s tensor. The elastic properties of an isotropic solid are rather simple. If stress is applied, all the strains can be worked out from the magnitude and direction of the stress and just two elastic moduli (the bulk modulus and the rigidity modulus). But in a crystal the elastic properties can vary considerably from one direction to another. Green showed that in the most general case 21 different moduli are needed to describe the strain. He also showed how symmetry can reduce this number. He became involved in this problem because he was interested in the "aether". At that time scientists believed that a real medium, the aether, existed everywhere. In outer space it was needed to carry vibrations of the light coming to us from the stars. Fresnel had shown that light was transverse wave so the aether had to be a solid (!) since gases and liquids could support only longitudinal waves. So Green started to analyse the properties of waves in solids and this brought him immediately to consider their elastic properties. He also calculated how much of a wave was reflected and how much was transmitted at an interface and explained the phenomenon of total internal reflection. In this work he was also the first to write down the principle of the conservation of energy, which had still to be established experimentally. His later work also contained a number of mathematical "firsts" including for example his work on hydrodynamics where he devised a powerful approximate method for solving differential equations which reappeared over a century later as the Wentzel-Kamers-Brillouin (WKB) method. He was also the first to state Dirichlet’s principle, although Riemann gave it the name it usually bears.
Recognition
We have seen that, during his life, very little recognition was paid to Green’s exceptional ability. His major work on electricity and magnetism was largely overlooked in Britain and unknown elsewhere. His contributions in other fields which were published beteen 1835 and 1839 were better known to his contemporaries but their true value was not appreciated until later. His standing is reflected by the lack of any obituary other than the grudging comment in a Nottingham paper which was quoted earlier.
Happily, though, that is not the end of the story. In the year Green died, William Thomson (later Lord Kelvin) went to Cambridge at the age of 17. He already had a mathematics degree from the University of Glasgow, where his father was Professor of Mathematics, and in the same year he published a paper on electrostatics. His interest in the theory of electricity continued and in 1845, just after he had taken his Cambridge degree, he arranged to go to Paris for a few months to work with French scientists active in this field. He had learned of Green’s essay but, not surprisingly, had been unable to find a copy in any of the bookshops. Fortunately, his tutor, Hopkins, had a copy which Thompson read on the way to Paris. His astonishment at what Green had accomplished is recalled in a letter he wrote to Sir Joseph Larmor shortly before his death in 1907. In it he describes very vividly how he read the essay on top of a stagecoach and how he showed it to the French mathematicians as soon as he arrived. They then discovered that Green had already done much of what they thought was their original work!
Thomson was greatly influenced by Green and adopted many of his techniques, as did Stokes. To quote Sir Edmund Whittaker’s authoritative history of Theories of Aether and Electricity – "It is no exaggeration to describe Green as the real founder of that "Cambridge school" of natural philosophers of whom Kelvin, Stokes, Lord Rayleigh and Clerk Maxwell were the most illustrious members in the latter half of the nineteenth century". The significance of his work on elasticity is described in Love’s Mathematical Theory of Elasticity which speaks of "the revolution which Green effected in the elements of the theory".
So practising scientists have no doubt of the importance of Green’s contribution. But what of the world outside? Even in Nottingham, Green was an obscure figure until recently, despite attempts in the 1920s to learn more about him and efforts in the 1930s by the British Association which led to his grave being restored. The brick tower of his windmill still stood in Sneinton, though it was a sad sight: it had been totally burned out following a fire in 1947 during a brief existence as a polish factory. The wooden sails and balcony had gone long before, although in 1921 the then owner, Oliver Hind, had clad the rotting wooden cap (roof) with copper to make it watertight. At about the same time, the Holbrook bequest had placed a plaque on the side of the mill to record its association with Green, although sadly this disappeared in 1969. The family house still stands and, thanks to the present owner, this also has a plaque.
In 1974, there was a rumour that the mill might be knocked down to make way for a by-pass.
The timing could not have been less appropriate. Green’s functions were being very widely used throughout the world, and this was hardly the moment for such destruction. So the George Green Memorial Fund was created with the aim of restoring the mill to provide a living and educational memorial to George Green. The City of Nottingham was immensely supportive of the idea and finally in July 1985 the mill was "opened" together with a Science Centre built on the foundations of the stables and storage areas. It is now grinding corn under the direction of the miller, David Bent. The Science Centre which was opened appropriately by Sir Sam Edwards, Cavendish Professor of Physics at Cambridge and Fellow of Gonville and Caius, contains a number of working models illustrating Green’s interest in electricity and magnetism, optics and elasticity. Green’s theorem is illustrated by a fountain sculpted following a national competition, by Mr Ron Haselden, and Green’s functions are presently illustrated by a computer game (although suggestions for a working exhibit would be gratefully received). The mill and science centre attract large numbers of visitors including school parties and is a splendid sight on the Nottingham skyline.
Many people and organisations have contributed to this project. It was effectively launched in 1974 by a telegram to the Lord Mayor of Nottingham from the 500 delegates at an International Physics Conference in Budapest and it was appropriate that a telegram of thanks could be sent from a similar conference, also in Budapest, in the autumn of 1985. The national scientific societies have been very supportive and the Institute of Physics can be singled out for its generous donation towards the scientific exhibits. Altogether, the George Green Memorial Fund raised nearly £40,000 towards the project, including the purchase of the mill in 1979, but the major contributor by far was the City of Nottingham. The success of these efforts and the publicity the project has received throughout the world has ensured that George Green has at last had the recognition he was so sadly denied in his lifetime.
Acknowledgements
The information on Green’s life comes with many thanks from the writings of the late Gwynneth Green, David Phillips, Freda Wilkins-Jones and Mary Cannell. I am also very grateful to my colleagues, particularly Fred Sheard, for discussions on Green’s contribution to science.
References and further information
Green’s Mill and Science Centre is open, without charge, from Tuesday to Sunday. Further details can be obtained from Mr D Plowman, Keeper, Green’s Mill and Science Centre, Belvoir Hill, Sneinton, Nottingham, NG2 4LF. Telephone (0115) 9503635.
Further information about Greens life may be found in George Green, Miller, Snienton, ed. D. Phillips (1976). It is now out of print but available in a number of libraries. A new book on Green by D. M. Cannell should be available within the next year. Further information about his mathematics appears in an article "The Work and Significance of George Green, the miller mathematician", by J. E. G. Farina, Bulletin of the Institute of Mathematics and its Applications 12 (1976), 98-105.
Facsimile editions of Green’s major essays may be purchased here.
This article is reproduced by permission of the Applied Probability Trust .from Mathematical Spectrum, 20(1987/8), 45-52
Copyright (c) The Applied Probability Trust 1987.
Reproduced with permission.
http://www.appliedprobability.org/
To avoid confusion, the telephone number of Green’s mill has been updated.
Lawrie Challis was Professor of Low Temperature Physics at the University of Nottingham. His main interests were experimental studies of electrons in semiconductors using very high frequency sound at low temperatures. He was also chairman of the George Green Memorial Fund which contributed to the restoration of Green’s windmill, now open to the public and grinding corn again.
Introduction
George Green (1793-1841) was a pioneer in the application of mathematics to physical problems. He was a miller who lived in Nottingham virtually all his life and had very little formal education until he had completed most of his best work. Partly as a result of his unusual circumstances he received little public recognition in his lifetime, and it was William Thomson (Lord Kelvin) who first recognised the value of his work and gave it wide publicity. His work has had great influence and nowadays he is remembered principally for Green’s theorem in vector analysis, Green’s tensor (or the Cauchy-Green tensor) in elasticity theory and above all for Green’s functions for solving differential equations. The Green’s function technique has been very widely applied to equations arising in classical physics and engineering and in recent years has been adapted to quantum mechanical problems in areas as diverse as nuclear physics, quantum electrodynamics and superconductivity. Yet until quite recently Green was a shadowy figure whose name was practically unknown to most non-scientists even in his home town. All this has now changed and, with the help of many organisations and of interested people from many countries, the City of Nottingham has created a splendid and living memorial to George Green. They have restored his ruined windmill to full working order and built a Science Centre illustrating some of his many interests.
George Green’s life
George Green was born in Nottingham on 13 July 1793. For generations his ancestors had been farmers in the village of Saxondale just a few miles from Nottingham, but his father, the youngest of three sons, had been sent there in 1774 to be apprenticed to a baker in Nottingham. In time he bought his own bakery and prospered, acquiring both land and property, which he rented out, as well as a warehouse on the banks of the River Lean which he used to store grain before sending it to be milled into flour for his bakery. When George was eight, he was sent to Robert Goodacre’s Academy. His schooling lasted only four terms, however, and then he left to help in the bakery. He was lucky in that his father sent him to that particular school, as Robert Goodacre was an enthusiastic science teacher. Indeed, later in his life, Goodacre became a popular speaker on astronomy, lecturing throughout the British Isles and America. So George Green would have acquired a taste for science, although it is doubtful whether his teacher could have stretched him very far in mathematics; Goodacre had had no formal training and had been apprenticed to a tailor before becoming a schoolmaster.
So at the age of nine, George Green had received all the formal education he was going to get until he was 40! There were of course bookshops in Nottingham where he could buy textbooks and encyclopedias, but there were as yet no libraries. It is possible that he may have received some guidance in his reading from one of the graduate mathematicians living in Nottingham.
When Green was 14, his father built a windmill at Sneinton, then a separate village a mile or so from Nottingham. It was a fine five-storey brick tower mill with stables for eight horses and storage for hay and corn. Milling was a skilled trade and he employed a foreman-manager, William Smith, who lived in a cottage attached to the side of the mill. The mill could not easily be worked single-handed; George helped William Smith, and so learned to operate the mill himself. This must have been an exciting change from the bakery for a boy of 14, and it would have been a strenuous, mostly outdoor life. When there was wind enough, he would have worked long hours even at night trimming the sails, maintaining the inflow of grain and taking away the milled flour or cattle fodder. As the Greens were still living in Nottingham, it seems likely that George might have stayed overnight with the Smiths or perhaps slept in the mill rather than walking back through the dark and probably dangerous streets of overcrowded Nottingham. It seems likely too that during this time he would have spent some of the calm days studying mathematics while waiting for the wind to come. Certainly his youngest daughter, Clara, who lived until 1919, told Professor Granger of University College, Nottingham, that her father used the top floor of the mill as a study.
When George was 24, he and his parents moved to a five-bedroomed house they built next to the mill and a few years later he joined the recently-opened Nottingham Subscription Library. This soon became the centre of intellectual life in Nottingham. It contained a modest collection of mathematical and scientific textbooks, and, of great importance, it took the important British scientific journals. These usually also included the titles and abstracts of papers from foreign journals, so that Green would have been able to follow what was being done elsewhere. In principle, he could then have written to the authors asking for copies of their papers.
In 1828, at the age of 35, George Green published his first paper "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism". It was a major work of striking originality. He invented completely new mathematical techniques to solve the problems that arose in the analysis and it would have had an immediate and profound effect had it been read by others working in the field. Tragically, it did not have this effect until some years after his death. He was advised that as he had had no formal training and his social position was modest, he could not presume to send his paper to a scientific journal! So instead he had it printed privately in Nottingham and presumably distributed a few copies to other mathematicians and physicists working in Britain. It had virtually no impact; hardly anyone in Britain had worked in this field for 50 years. British mathematicians were interested in mechanics, optics, astronomy, planetary motion and hydrodynamics; Green’s inspiration came from France, from Laplace and Poisson, but nobody there appears to have seen his paper. This lack of response must have depressed Green, but he soon started work on a second paper. He received valuable encouragement from Sir Edward Bromhead, an influential Cambridge mathematics graduate who lived in Lincolnshire and clearly realised Green’s exceptional ability.
Green turned to areas of much more interest to British mathematical physicists and, with Bromhead’s connections as his passport, he started to publish papers in the scientific journals. His family life also changed considerably about this time, when his father died. His mother had died some years before, so Green became a rather wealthy man. He stopped milling, leasing out his mill, and then in 1833, at the age of 40, he realised what may have been his ambition for many years. With Bromhead’s help he entered Cambridge as an undergraduate to read for a degree in mathematics. He took his degree in 1837 and shortly afterwards was elected to a fellowship at his college, Gonville and Caius.
Sadly, he held the post for only two years before becoming ill and returning to Nottingham, where he died in 1841. Sadly too, the full value of his work was not appreciated until after his death. Indeed, the only obituary that appeared was in a local Nottingham paper. It was three sentences long and ends "Had his life been prolonged, he might have stood eminently high as a mathematician".
Green’s mathematics
Green’s mathematics was nearly all devised to solve very general physical problems. His first interest was in electrostatics. The inverse-square law had recently been established experimentally, and he wanted to calculate how this determined the distribution of charge on the surfaces of conductors. He made great use of the electrical potential, and gave it that name, and one of the theorems that he proved in this work became famous and is known as Green’s theorem. It relates the properties of mathematical functions at the surfaces of a closed volume to other properties inside. In its usual form, the theorem involves two functions, but it readily simplifies to what is often called the divergence theorem or Gauss’s theorem. (Many early textbooks also called this form Green’s theorem as well, presumably to emphasise his claims to precedence).
To illustrate the theorem, we consider gas leaking from holes in the walls of a gas cylinder. The mass leaving per second per unit area equals the product of the density of the gas and its velocity at each hole. So we can find the total loss rate by integrating this over all the holes. (The integral can in fact be carried out over the whole surface since the contribution from the rest is zero). But this loss rate from the surface must equal the sum of the masses leaving per second from all the small volume elements dV inside the surface and this can be found by integrating a particular function over the whole volume V. The function is the result of a differential operator 4 called the divergence acting on the product of density and velocity of the gas at the element dV. The theorem relating the integral over the surface to the integral over the volume inside is useful in many branches of physics. For example, in electrostatics, a development closely related to this links the electric flux leaving a surface to the total charge inside it.
Another powerful technique invented by Green is used for solving differential equations. It involves what are now called Green’s functions, G(x,x’). If we have a differential equation Ly = F(x), where L is a linear differential operator, then the solution can be written
To see this, consider the equation
This can be solved by the standard integrating factor technique to give
so that G(x,x’) = exp[-k(x-x')].
To understand the meaning of this in a physical situation, consider the motion of a unit mass (initially at rest) following the application of a time-varying force F(t). If the motion is damped by a force –kv then, from Newton’s law,
with solution
as before. This can be interpreted by visualising the time-varying force F(t) as a rapid sequence of sharp blows each acting for a short time dt’ and causing a change in momentum or impulse F(t’)dt’. Thus the velocity v(t) at time t is the sum (or integral) of the effects of all blows from t’ = 0 to t’ = t. The velocity at time t due to a single impulse (with unit change of momentum) applied at time t’ is called Green’s function
This technique may be applied to other more complicated systems. In an electrical circuit the Green’s function is the current due to an applied voltage pulse. In electrostatics the Green’s function is the potential due to a change applied at a particular point in space. In general the Green’s function is the response of a system to a stimulus applied at a particular point in space or time. This concept has been readily adapted to quantum physics where the applied stimulus is the injection of a quantum of energy. It is in the quantum domain that the application of Green’s functions to physical problems has grown most spectacularly in the past few decades.
Green also did very original work on elasticity, where he is remembered by Green’s tensor. The elastic properties of an isotropic solid are rather simple. If stress is applied, all the strains can be worked out from the magnitude and direction of the stress and just two elastic moduli (the bulk modulus and the rigidity modulus). But in a crystal the elastic properties can vary considerably from one direction to another. Green showed that in the most general case 21 different moduli are needed to describe the strain. He also showed how symmetry can reduce this number. He became involved in this problem because he was interested in the "aether". At that time scientists believed that a real medium, the aether, existed everywhere. In outer space it was needed to carry vibrations of the light coming to us from the stars. Fresnel had shown that light was transverse wave so the aether had to be a solid (!) since gases and liquids could support only longitudinal waves. So Green started to analyse the properties of waves in solids and this brought him immediately to consider their elastic properties. He also calculated how much of a wave was reflected and how much was transmitted at an interface and explained the phenomenon of total internal reflection. In this work he was also the first to write down the principle of the conservation of energy, which had still to be established experimentally. His later work also contained a number of mathematical "firsts" including for example his work on hydrodynamics where he devised a powerful approximate method for solving differential equations which reappeared over a century later as the Wentzel-Kamers-Brillouin (WKB) method. He was also the first to state Dirichlet’s principle, although Riemann gave it the name it usually bears.
Recognition
We have seen that, during his life, very little recognition was paid to Green’s exceptional ability. His major work on electricity and magnetism was largely overlooked in Britain and unknown elsewhere. His contributions in other fields which were published beteen 1835 and 1839 were better known to his contemporaries but their true value was not appreciated until later. His standing is reflected by the lack of any obituary other than the grudging comment in a Nottingham paper which was quoted earlier.
Happily, though, that is not the end of the story. In the year Green died, William Thomson (later Lord Kelvin) went to Cambridge at the age of 17. He already had a mathematics degree from the University of Glasgow, where his father was Professor of Mathematics, and in the same year he published a paper on electrostatics. His interest in the theory of electricity continued and in 1845, just after he had taken his Cambridge degree, he arranged to go to Paris for a few months to work with French scientists active in this field. He had learned of Green’s essay but, not surprisingly, had been unable to find a copy in any of the bookshops. Fortunately, his tutor, Hopkins, had a copy which Thompson read on the way to Paris. His astonishment at what Green had accomplished is recalled in a letter he wrote to Sir Joseph Larmor shortly before his death in 1907. In it he describes very vividly how he read the essay on top of a stagecoach and how he showed it to the French mathematicians as soon as he arrived. They then discovered that Green had already done much of what they thought was their original work!
Thomson was greatly influenced by Green and adopted many of his techniques, as did Stokes. To quote Sir Edmund Whittaker’s authoritative history of Theories of Aether and Electricity – "It is no exaggeration to describe Green as the real founder of that "Cambridge school" of natural philosophers of whom Kelvin, Stokes, Lord Rayleigh and Clerk Maxwell were the most illustrious members in the latter half of the nineteenth century". The significance of his work on elasticity is described in Love’s Mathematical Theory of Elasticity which speaks of "the revolution which Green effected in the elements of the theory".
So practising scientists have no doubt of the importance of Green’s contribution. But what of the world outside? Even in Nottingham, Green was an obscure figure until recently, despite attempts in the 1920s to learn more about him and efforts in the 1930s by the British Association which led to his grave being restored. The brick tower of his windmill still stood in Sneinton, though it was a sad sight: it had been totally burned out following a fire in 1947 during a brief existence as a polish factory. The wooden sails and balcony had gone long before, although in 1921 the then owner, Oliver Hind, had clad the rotting wooden cap (roof) with copper to make it watertight. At about the same time, the Holbrook bequest had placed a plaque on the side of the mill to record its association with Green, although sadly this disappeared in 1969. The family house still stands and, thanks to the present owner, this also has a plaque.
In 1974, there was a rumour that the mill might be knocked down to make way for a by-pass.
The timing could not have been less appropriate. Green’s functions were being very widely used throughout the world, and this was hardly the moment for such destruction. So the George Green Memorial Fund was created with the aim of restoring the mill to provide a living and educational memorial to George Green. The City of Nottingham was immensely supportive of the idea and finally in July 1985 the mill was "opened" together with a Science Centre built on the foundations of the stables and storage areas. It is now grinding corn under the direction of the miller, David Bent. The Science Centre which was opened appropriately by Sir Sam Edwards, Cavendish Professor of Physics at Cambridge and Fellow of Gonville and Caius, contains a number of working models illustrating Green’s interest in electricity and magnetism, optics and elasticity. Green’s theorem is illustrated by a fountain sculpted following a national competition, by Mr Ron Haselden, and Green’s functions are presently illustrated by a computer game (although suggestions for a working exhibit would be gratefully received). The mill and science centre attract large numbers of visitors including school parties and is a splendid sight on the Nottingham skyline.
Many people and organisations have contributed to this project. It was effectively launched in 1974 by a telegram to the Lord Mayor of Nottingham from the 500 delegates at an International Physics Conference in Budapest and it was appropriate that a telegram of thanks could be sent from a similar conference, also in Budapest, in the autumn of 1985. The national scientific societies have been very supportive and the Institute of Physics can be singled out for its generous donation towards the scientific exhibits. Altogether, the George Green Memorial Fund raised nearly £40,000 towards the project, including the purchase of the mill in 1979, but the major contributor by far was the City of Nottingham. The success of these efforts and the publicity the project has received throughout the world has ensured that George Green has at last had the recognition he was so sadly denied in his lifetime.
Acknowledgements
The information on Green’s life comes with many thanks from the writings of the late Gwynneth Green, David Phillips, Freda Wilkins-Jones and Mary Cannell. I am also very grateful to my colleagues, particularly Fred Sheard, for discussions on Green’s contribution to science.
References and further information
Green’s Mill and Science Centre is open, without charge, from Tuesday to Sunday. Further details can be obtained from Mr D Plowman, Keeper, Green’s Mill and Science Centre, Belvoir Hill, Sneinton, Nottingham, NG2 4LF. Telephone (0115) 9503635.
Further information about Greens life may be found in George Green, Miller, Snienton, ed. D. Phillips (1976). It is now out of print but available in a number of libraries. A new book on Green by D. M. Cannell should be available within the next year. Further information about his mathematics appears in an article "The Work and Significance of George Green, the miller mathematician", by J. E. G. Farina, Bulletin of the Institute of Mathematics and its Applications 12 (1976), 98-105.
Facsimile editions of Green’s major essays may be purchased here.
Thursday, 31 May 2007
George Green
The Green Function was invented by George Green (1793-1841), one of the most remarkable of nineteenth century physicists, a self-taught mathematician whose work has contributed greatly to modern physics.
For information on his life and work see the pages at the BA web site.
For information on his life and work see the pages at the BA web site.
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