Lagrange was extremely active scientifically during the twenty years he spent in Berlin. Not only did he produce his
Mécanique analytique, but he contributed between one and two hundred papers to the Academy of Turin, the Berlin Academy, and the French Academy. Some of these are really treatises, and all without exception are of a high order of excellence. Except for a short time when he was ill he produced on average about one paper a month. Of these, note the following as amongst the most important. First, his contributions to the fourth and fifth volumes, 1766–1773, of the
Miscellanea Taurinensia; of which the most important was the one in 1771, in which he discussed how numerous
astronomical observations should be combined so as to give the most probable result. And later, his contributions to the first two volumes, 1784–1785, of the transactions of the Turin Academy; to the first of which he contributed a paper on the pressure exerted by fluids in motion, and to the second an article on integration by
infinite series, and the kind of problems for which it is suitable. Most of the papers sent to Paris were on astronomical questions, and among these, including his paper on the
Jovian system in 1766, his essay on the problem of three bodies in 1772, his work on the
secular equation of the Moon in 1773, and his treatise on cometary perturbations in 1778. These were all written on subjects proposed by the
Académie française, and in each case, the prize was awarded to him.
Lagrangian mechanics Between 1772 and 1788, Lagrange re-formulated Classical/Newtonian mechanics to simplify formulas and ease calculations. These mechanics are called
Lagrangian mechanics.
Algebra The greater number of his papers during this time were, however, contributed to the
Prussian Academy of Sciences. Several of them deal with questions in
algebra. • His discussion of representations of integers by
quadratic forms (1769) and by more general algebraic forms (1770). • His tract on the
Theory of Elimination, 1770. •
Lagrange's theorem that the order of a subgroup H of a group G must divide the order of G. • His papers of 1770 and 1771 on the general process for solving an
algebraic equation of any degree via the
Lagrange resolvents. This method fails to give a general formula for solutions of an equation of degree five and higher because the auxiliary equation involved has a higher degree than the original one. The significance of this method is that it exhibits the already known formulas for solving equations of second, third, and fourth degrees as manifestations of a single principle, and was foundational in
Galois theory. The complete solution of a binomial equation (namely an equation of the form ax^n ± b=0) is also treated in these papers. • In 1773, Lagrange considered a
functional determinant of order 3, a special case of a
Jacobian. He also proved the expression for the
volume of a
tetrahedron with one of the vertices at the origin as the one-sixth of the
absolute value of the
determinant formed by the coordinates of the other three vertices.
Number theory Several of his early papers also deal with questions of number theory. • Lagrange (1766–1769) was the first European to prove that
Pell's equation has a nontrivial solution in the integers for any non-square natural number . • He proved the theorem, stated by
Bachet without justification, that
every positive integer is the sum of four squares, 1770. • He proved
Wilson's theorem that (for any integer ): is a prime if and only if is a multiple of , 1771. • His papers of 1773, 1775, and 1777 gave demonstrations of several results enunciated by Fermat, and not previously proved. • His
Recherches d'Arithmétique of 1775 developed a general theory of binary
quadratic forms to handle the general problem of when an integer is representable by the form . • He made contributions to the theory of
continued fractions.
Other mathematical work There are also numerous articles on various points of
analytical geometry. In two of them, written rather later, in 1792 and 1793, he reduced the
equations of the quadrics (or conicoids) to their
canonical forms. During the years from 1772 to 1785, he contributed a long series of papers which created the science of
partial differential equations. A large part of these results was collected in the second edition of Euler's integral calculus which was published in 1794.
Astronomy Lastly, there are numerous papers on problems in
astronomy. Of these the most important are the following: • Attempting to solve the
general three-body problem, with the consequent discovery of the two constant-pattern solutions, collinear and equilateral, 1772. Those solutions were later seen to explain what are now known as the
Lagrangian points. • On the attraction of ellipsoids, 1773: this is founded on
Maclaurin's work. • On the secular equation of the Moon, 1773; also noticeable for the earliest introduction of the idea of the potential. The potential of a body at any point is the sum of the mass of every element of the body when divided by its distance from the point. Lagrange showed that if the potential of a body at an external point were known, the attraction in any direction could be at once found. The theory of the potential was elaborated in a paper sent to Berlin in 1777. • On the motion of the nodes of a planet's
orbit, 1774. • On the stability of the planetary orbits, 1776. • Two papers in which the method of determining the orbit of a
comet from three observations is completely worked out, 1778 and 1783: this has not indeed proved practically available, but his system of calculating the perturbations by means of mechanical quadratures has formed the basis of most subsequent researches on the subject. • His determination of the secular and periodic variations of the
elements of the planets, 1781–1784: the upper limits assigned for these agree closely with those obtained later by
Le Verrier, and Lagrange proceeded as far as the knowledge then possessed of the masses of the planets permitted. • Three papers on the method of interpolation, 1783, 1792 and 1793: the part of finite differences dealing therewith is now in the same stage as that in which Lagrange left it.
Fundamental treatise Over and above these various papers he composed his fundamental treatise, the
Mécanique analytique. In this book, he lays down the law of virtual work, and from that one fundamental principle, by the aid of the calculus of variations, deduces the whole of
mechanics, both of solids and fluids. The object of the book is to show that the subject is implicitly included in a single principle, and to give general formulae from which any particular result can be obtained. The method of generalised co-ordinates by which he obtained this result is perhaps the most brilliant result of his analysis. Instead of following the motion of each individual part of a material system, as D'Alembert and Euler had done, he showed that, if we determine its configuration by a sufficient number of variables
x, called
generalized coordinates, whose number is the same as that of the degrees of freedom possessed by the system, then the kinetic and potential energies of the system can be expressed in terms of those variables, and the differential equations of motion thence deduced by simple differentiation. For example, in dynamics of a rigid system he replaces the consideration of the particular problem by the general equation, which is now usually written in the form : \frac{d}{dt} \frac{\partial T}{\partial \dot{x}} - \frac{\partial T}{\partial x} + \frac{\partial V}{\partial x} = 0, where
T represents the kinetic energy and
V represents the potential energy of the system. He then presented what we now know as the method of
Lagrange multipliers—though this is not the first time that method was published—as a means to solve this equation. Amongst other minor theorems here given it may suffice to mention the proposition that the kinetic energy imparted by the given impulses to a material system under given constraints is a maximum, and the
principle of least action. All the analysis is so elegant that Sir
William Rowan Hamilton said the work could be described only as a scientific poem. Lagrange remarked that mechanics was really a branch of
pure mathematics analogous to a geometry of four dimensions, namely, the time and the three coordinates of the point in space; and it is said that he prided himself that from the beginning to the end of the work there was not a single diagram. At first no printer could be found who would publish the book; but
Legendre at last persuaded a Paris firm to undertake it, and it was issued under the supervision of Laplace, Cousin, Legendre (editor) and Condorcet in 1788. ==Work in France==