2011 Fall Math 5305 Ramiro Garza
Lagrange
Ramiro Garza 1
Joseph-Louis Lagrange was born
Guiseppe Lodovico (Luigi) Lagrangia in
Turin, Piedmont, Italy. He was the first of eleven children, but only he and one other child made it to adulthood. He was born with an Italian name, to Italian parents, inside the Italian country. However, he did have French lineage on his father's side, and seemed to prefer that personal history, even from a young age, as he would often sign his name Lagrange. The better part of his life was lived in Prussia and France.
For this reason, he is considered a French and Italian scientist. His significant contributions were to all fields of analysis, number theory, and classical and celestial mechanics.
Figure 1: Lagrange [16]
His father had charge of the
Kingdom of Sardinia’s military chest, a position that benefited both his social and financial situations. However, because his father did not know how to invest well, most of the wealth was gone by the time Lagrange grew up. For this reason,
Lagrange had to make his own way in the world. His father reared him to become a lawyer, and rather liked the prospect. While studying at the college of Turin, already on the legal path, Lagrange was doing quite well, and his favorite subject was Classical Latin. In fact, he actually found the math he was studying to be insipid.
He reportedly said, “If I had been rich, I probably would not have devoted myself to mathematic s.” He didn’t get turned on to mathematics until he was about seventeen, after a happy accident that led to him reading a paper by
Edmund Halley. After only a year of relentless work, he made himself an accomplished mathematician and was made a lecturer in the artillery school.
In 1758, with the aid of his pupils, Lagrange established a society that was eventually made a part of the Turin Academy of Sciences. Most of his early work is in the transactions of this society, consists of five volumes, and is usually called Miscellanea Taurinensia .
1 University of Texas at Brownsville, 28 October 2011
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2011 Fall Math 5305 Ramiro Garza
In the first volume is a paper of his that indicates a mistake in the study of the propagation of sound made by
Newton. He corrects it and develops the study further, and also gives the complete solution of a string vibrating transversely, such as a guitar string. He points out the lack of generality of previous solutions to this problem, given by
Brook Taylor, D’Alembert and Euler.
His conclusion is that the form of the curve of such a string takes the form of 𝑦 = 𝑎 sin(𝑚𝑥) sin(𝑛𝑡) . He continues with the discussion of echoes, beats and compound sounds. Other articles that can be found in the first volume are on recurring series, probabilities and the calculus of variations.
Figure 2: Soap, an example of variational calculus [24]
The calculus of variations is a field of calculus that Lagrange created when he discovered a way to find the local maxima and minima of functionals that was very similar to the extrema method of functions. Lagrange wrote several letters to
Euler from 1754 and 1756 describing his work. He showed Euler what he called a 𝛿 -algorithm, and that led to the Euler-Lagrange equations of variational calculus. Additionally, this simplified Euler’s earlier analysis and Lagrange was able to apply these ideas of classical mechanics, generalizing earlier results that had been obtained.
It is said that Euler was not only impressed with Lagrange’s work, but was so impressed and courteous as so allow Lagrange the time to finish and publish the variational calculus work, and withheld his own research that had covered similar areas to do so. This would give Lagrange undisputed credit for the invention of this new calculus. However, this is not unanimously accepted as the truth. Lagrange eventually did publish the work with the Turin society in 1762 and
1773.
The second volume has in it several of his results based on the work and idea in the first volume, such as variational calculus. He describes the principle of least action, which he derives from his variational calculus methods, and when it is applied to the action of a mechanical system, the equations of motion can be obtained. He also published several solutions to various problems in dynamics, which is the study of forces and the objects affected into and out of motion by them.
The third volume builds on the first two, giving even more solutions to problems in dynamics through the application of variatonal calculus. He was the first person to give a full proof of a theorem stated by John Pell
—or so Euler
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2011 Fall Math 5305 Ramiro Garza though, but it seems that he confused Pell with Lord Brouncker
—and it is that, given a natural number, 𝑛 which is equal to no integer squared, there exists an integer, 𝑥 , such that 𝑥 2 𝑛 + 1 is equal to some other integer, 𝑦 , squared. The ratio of 𝑦/𝑥 yields good approximations for the square root of 𝑛 .
He also published the general differential equations of the motion for three bodies moving under their mutual attractions. This was typically understood to be celestial bodies, like planets or stars, moving under the mutual attraction of gravity. Specific solutions to this problem yield examples of chaotic motion, where there is no sign of pattern.
In 1756, Euler, with the support of
Maupertius, attempted to bring Lagrange to the
Berlin Academy. Later,
D’Alembert and King
Frederick of Prussia also called him to the Berlin
Academy. His response was that, “It seems to me that Berlin would not be at all suitable for me while Monsieur Euler is there.” When Euler finally
Figure 3: A modeling of the three-body problem [26} did leave, in 1766 for
Saint Petersburg,
Frederick wrote to L agrange again, expressing the wish of “the greatest king in
Europe” to have “the greatest mathematician in Europe.” Lagrange was finally convinced to move, and he stayed in Prussia for the next twenty years. Upon arrival in Berlin, Lagrange came to find that most of his colleagues were married, and they, along with their wives, convinced Lagrange that it was the only path to happiness. He heeded their advice and was soon married, reportedly to a cousin of his, which was a complete disaster. They were not happy together, and his wife passed away not long after their matrimony.
Not only did he continue to publish quality papers for Berlin and Turin academies, but also it was here that he wrote his magnum opus, his Mécanique analytique . MA is a grand treatise, written in two volumes, which offered the most comprehensive treatment of classical mechanics since Newton. Much of his work in the physics of mechanics is, actually, a revision of Newton’s work, reworked to simplify calculations. Lag range’s formulation identifies the path followed by motion by selecting the path in which the integral, over time, of kinetic energy is minimized. MA developed the mathematical physics in the nineteenth century.
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Lagrange came to be a favorite of King Frederick, with them having occasional talks. Some of what the king had to say to Lagrange inspired him to challenge himself every day with definite goals in mind to be done by day’s end.
He pushed himself to his maximum abilities, both physically and mentally. He worked himself until he found his breaking point. Lagrange examined all areas of science that interested him and investigated what he could do to further or improve these fields.
When Frederick died in 1786, Lagrange willingly accepted King Louis
XVI’s invitation to France. Lagrange had found the climate of Berlin unpleasant and was happy to move to Paris. Offers also came to him from Spain and Naples.
In France, he received all marks of distinction and special apartments in the
Louvre were prepared for him. He became a member of the French Academy of
Sciences, an academy that was eventually incorporated into the National Institute.
Upon first arriving in
France, a sudden pensive depression hit him, with no real traceable reason. One of the first perturbations to stir him from this melancholy were the goings-on of the
French Revolution. He was, at first, curious as the situation began. Soon, however, the reality of the situation and its seriousness brought him to fear and serious consideration of fleeing France.
Figure 4: Fluid mechanics [27]
Lagrange was a timid man and this, along with the depression that was consuming him, was enough to make a young girl insist upon marrying him. Over their years together, she proved to be a devoted with and they became warmly attached.
Now, Lagrange was still, and more seriously, planning escape from
France
—despite the fact that the decree of October 1793 that ordered all nonnative persons to leave France specifically stated his name as an exemption
— but then he was offered the presidency of the commission for the reform of weights and measures. His great influence in the group is one of the main reasons for the specific units chosen in 1799. The units were additionally subdivided according to a decimal system; again, largely due to his insistence and the influence he had in the decision-making process.
On a related note, Lagrange became one of the founding members of the
Bureau des Longitudes. The Bureau was created to increase the uniformity of measure and accuracy of methods used in France. The main cause for this
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2011 Fall Math 5305 Ramiro Garza formation was the British control over the seas, which was due to their advanced navy, and was the greatest contributor to their power. It was believed that, with greater accuracy of location, the French naval force could be brought to equal footing with that of Great Britain, and that the French could reclaim some of the control they had lost.
Lagrange still sought to leave France before it was too late, but history will show that he was never in any real danger. Not only did the various powers that came to be during this tumultuous period not want to harm him, they bestowed upon him further honors. He was a highly respected and valued person, and none of these revolutionaries were foolish enough to believe otherwise. Even
Napoleon honored him, encouraged furthering studies in all sciences, as well as liberally investing money in their research.
1795 saw the establishment of the École normale, but it only existed for a brief for four months. For this time, Lagrange was appointed a mathematical chair and worked as a lecturer. The lectures he gave were elementary for the most part, but all lectures were ordered to be recorded in shorthand for legal reasons. Lagran ge served as a professor of the École Polytechnique, and, those mathematicians who were lucky enough to be there described them as being perfect in form and matter. Fourier wrote that the students were incapable of truly appreciating the man, but that the faculty “made amends for it.”
In his later years, beginning in 1810, Lagrange set out to revise his
Mécanique analytique , but, when he passed away in 1813, he had only gotten through about twothirds of the work. He was buried in the Panthéon in France, an exceptional building of structure and beauty, which became a mausoleum for
France’s most reputable inhabitants. The inscription on his tomb reads:
JOSEPH LOUIS LAGRANGE. Senator. Count of the Empire. Grand
Officer of the Legion of Honour. Grange Cross of the Imperial Order of Réunion. Member of the Institute and the Bureau of Longitude.
Born in Turin on 25 January 1736. Died in Paris on 10 April 1813.
Lagrange almost never sat still, as far as work went. While in Berlin, he produced between one and two hundred papers to the Academy of Turin, the
Berlin Academy and the French Academy. Each paper is described to be on a high order of excellence. Aside from when he was ill, Lagrange produced, on average, one paper per month. In 1771, he produced a paper for the Miscellanea
Taurinensia , in which he discussed how numerous astronomical observations should be combined to give the most probable result (such as location). For the
Turin Academy, he produced papers on the pressure exerted by fluids in motion, as well as one on the integration by infinite series, including the sorts of problems to which the method could be applied.
Most of the papers that went to the French academy were in the realm of the heavens. He wrote on the Jovian system in 1766, the Moon in 1773, comet-
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2011 Fall Math 5305 Ramiro Garza related perturbations1778, and all were proposed by the Académie française. In each case, he was awarded the prize.
The bulk of his papers, whilst in Berlin, went to the Prussian Academy of
Sciences. Several were on the subject matter of algebra. He discussed the representation of integers by quadratic forms, which meant representing them with polynomials in which all the terms had the same additive degree of two
(multiple variables, e.g.: 4𝑥 2 + 2𝑥𝑦 − 3𝑦 2
). He wrote on the Theory of Elimination in 1770. He provided his theorem that the order of a subgroup, 𝐻 , of a group, 𝐺 , must divide the order of 𝐺 . In 1773, Lagrange considered a functional determinant of order three, a special case of the Jacobian. He also provided the expression for the volume of a tetrahedron with one of the vertices at the origin as one sixth of the absolute value of the determinant formed by the coordinates of the other three vertices.
In number theory, he proved Bachet’s conjecture that every positive integer is the sum of four squares in 1770. He proved Wilson’s theorem that, if 𝑛 is a prime, then (𝑛 − 1)! + 1 is always a multiple of 𝑛 in 1771. His papers in 1773,
1775 and 1777 gave demonstrations of several results enunciated by
Fermat, which, of course, he never bothered to prove.
Beginning in 1772 and working until 1785, Lagrange produced a long series of papers that created the science of partial differential equations.
A great part of these results were included in the second edition of
Euler’s integral calculus in 1794. Figure 5: A continued fraction approximating the square root of 2 [25]
He is considered a scientist rather than simply a mathematician due to the variety of sciences in which he worked, including math, physics and astronomy (although one could argue that the latter two are simply applied math problems of analysis). He wrote on the three-body problem (without solving it); the attraction of ellipsoids; the secular equation of the Moon; two papers which document the method of determining the orbit of a comet using only three observations (it is been deemed not practically available, but the system he used formed the basis for further research on the subject); three papers on interpolation, with the part of his work that dealt with finite differences has not been improved upon.
The next, and final, great portion of his life was lived in France. Among other things, he wrote on the infinitesimals, theoretical numbers that are situated between zero and all positive real numbers. He reverted to their use in preference to founding the differential calculus on the study of algebraic forms. In the second edition of
Mécanique analytique
, the justifies his use of infinitesimal and concludes that:
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When we have grasped the spirit of the infinitesimal method, and have verified the exactness of its results either by the geometrical method of prime and ultimate ratios, or by the analytical method of derived functions, we may employ infinitely small quantities as a sure and valuable means of shortening and simplifying our proofs.
On his work with continued fractions, he gives a method for approximating a square root using them, along with several other theorems. In a note at the end of one of his papers, he shows how Fermat’s little theorem: 𝑎 𝑝−1 − 1 = 0(𝑚𝑜𝑑 𝑝) , where 𝑝 is a prime and 𝑎 is prime to 𝑝 , may be applied to give the complete algebraic solution of any binomial equation.
The theory of planetary motions had been the subject of one of his more remarkable papers submitted to Berlin. In 1806, Poisson reopened the subject by reading it aloud to the French Academy. Poisson determined that Lagrange’s formulae led to certain limits for the stability of orbits. Lagrange was present and discussed the entire subject afresh. In a letter communicated to the Academy in
1808, Lagrange discussed how, by varying arbitrary constants, one could determine the periodical and secular inequalities of any system of mutually interacting bodies.
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2011 Fall Math 5305 Ramiro Garza
Euler proposed that Lagrange succeed him at the Berlin Academy and he
Figure 6: Fluid mechanics, more analytical [28] was elected on 2 September 1756. Among his other honors are and election as a
Fellow of the Royal Society of Edinburgh, appointment by Napoleon as a Grand
Officer of the Legion of Honour and a Comte of the Empire, prizes from the
French Academy of Sciences, and he is one of the 72 scientists whose names are engraved on the Eiffel Rower. Eiffel added these 72 names in his plans due to how he felt, that their contributions to science were invaluable to his work on the tower.
Lagrange gave the first known proof that all positive integers who were not perfect squares could be represented as the sum of four squares of integers.
This proof, although it deals with such basic operations and numbers, requires the use of very advanced numbers and methods. The numbers employed are quaternions; more specifically, Hurwitz quaternions.
Quaternions are an extension of the complex numbers, 𝑎 + 𝑏𝑖 , but are of the form 𝑎 + 𝑏𝑖 + 𝑐𝑗 + 𝑑𝑘 , where 𝑎, 𝑏, 𝑐, 𝑑 are real numbers. 𝑖 2 = 𝑗 2 = 𝑘 2 = 𝑖𝑗𝑘 =
−1.
For Herwitz quaternions, 𝑎, 𝑏, 𝑐, and 𝑑 are either all integers, or all halfintegers (where the only acceptable halves are an odd number divided by two). It is important to note that the norm of a quaternion, which is the product of the quaternion with its negative, save 𝑎 . This gives a norm equal to the sum of the squares of the coefficients, which is, therefore, always a nonnegative real number. It can be shown directly that the norm of a product of quaternions is the product of their norms.
Because all integers can be factored into a product of primes, and because of Euler’s four-square identity, the proof need only cover all primes, and then all numbers are covered. Euler’s identity shows that the sum of four squares times another sum of four squares equal a sum of four squares, where the number being squared is always an integer. Each prime can be made into a sum of four squares, and the product between primes’ sums of squares is proven to
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2011 Fall Math 5305 Ramiro Garza be another sum of squares. Thus, we have proof that we only need the proof for primes.
2 is the only even prime, and 2 = 1 2 + 1 2 + 0 2 + 0 2
. Then, we only need the proof for all odd primes. Any number, specifically any prime number, can be represented as a Hurwitz quaternion, where 𝑎 is equal to the number and 𝑏 = 𝑐 = 𝑑 = 0 .
Hurwitz quaternions form a ring, which means that they are closed with respect to addition and multiplication. Assume that an odd prime is not the product of two Hurwitz quaternions. That would make it, by definition, a Hurwitz prime. However, Lagrange proved that all odd primes divide at least one number of the form 1 + 𝑙 2 + 𝑚 2
, where 𝑙 and 𝑚 are integers. This number can be factored into two Hurwitz quaternions, namely (1 + 𝑙𝑖 + 𝑚𝑗)(1 − 𝑙𝑖 − 𝑚𝑗) . Because the prime divides 1 + 𝑙 2 + 𝑚 2 = (1 + 𝑙𝑖 + 𝑚𝑗)(1 − 𝑙𝑖 − 𝑚𝑗) , then it divides one of its factors. However, neither of the two quotients produced would be a Hurwitz quaternion. (I’m not sure why, but) this proves that all odd primes are divisible by two Hurwitz quaternions.
Thus, because any odd prime can be represented as the product of two
Hurwitz quaternions, the norm of it, equal to itself squared, is equal to the product of the two factors. Because this norm must be a real number, and, more specifically, a positive integer (even if the coefficients are halves, the norm will be a whole number because each pair of coefficients squared times four is exactly divisible once by two, then the two pairs are now odd again (times two), and all four can be then added up to an even number, which is divisible by two, and the factor of one-fourth has been eliminated), it has a prime factorization. Well, this factorization is precisely the prime number squared. Thus, the prime number is equal to the norm of each Hurwitz factor, which are, themselves, a sum of integers squared. (I’m not sure what the situation is when both of the factors are half-quaternions, and could not find a justification anywhere.) Quad erat demonstandum.
References:
[1] http://en.wikipedia.org/wiki/Lagrange
[2] http://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem
[3] http://en.wikipedia.org/wiki/Hurwitz_quaternion
[4] http://planetmath.org/encyclopedia/ProofOfLagrangesFourSquare
Theorem.html
[5] http://en.wikipedia.org/wiki/Calculus_of_variations
[6] http://en.wikipedia.org/wiki/Dynamics_(mechanics)
[7] http://en.wikipedia.org/wiki/Principle_of_least_action
[8] http://en.wikipedia.org/wiki/Lagrangian
[9] http://en.wikipedia.org/wiki/Euler-Lagrange
[10] http://en.wikipedia.org/wiki/N-body_problem
[11] http://www.gap-system.org/~history/Biographies/Lagrange.html
[12] http://en.wikipedia.org/wiki/French_revolution
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[13] http://en.wikipedia.org/wiki/Decimal_time
[14] http://headoverflow.com/decimal-clock/
[15] http://en.wikipedia.org/wiki/Bureau_des_Longitudes
[16] http://upload.wikimedia.org/wikipedia/commons/d/d8/Langrange_ portrait.jpg
[17] http://flatrock.org.nz/topics/working/assets/empty_nursery1.jpg
[18] http://blog.georgetownvoice.com/wpcontent/uploads/2010/09/ducktales.jpg
[19] http://droitanglais.org/wp-content/uploads/2011/05/Harvard-Law-
School.jpeg
[20] http://3.bp.blogspot.com/_XLVZ0IbIMas/TBSbQqziRQI/AAAAAAAACq0/
CddxI_mckdM/s1600/confused2.jpg
[21] http://www.bathsheba.com/sculpt/ora/ora_2.jpg
[22] http://www.metlin.org/content/blog/math/bathsheba-soliton_end.jpg
[23] http://upload.wikimedia.org/wikipedia/commons/2/2f/Clock-frenchrepublic.jpg
[24] http://math.unifr.ch/figs/analyse.jpg
[25] http://wiki-images.enotes.com/math/8/2/8/82865ae263d142ccf
4e6602a0366f666.png
[26] http://sprott.physics.wisc.edu/chaos/3body.gif
[27] http://www.master.sdi.upmc.fr/contributor/resources/download/ gestionsdi/Mecanique/M2/M2-Mecanique_fluides/Images/fig-atomphoto.gif
[28] http://www.ii.uib.no/vis/publications/publication/2010/images/ fuchs10lagrangian2.jpg
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