Carl Jacobi's British Association lecture
In June 1842 Carl Jacobi addressed the British Association at its 12th meeting, held in Manchester, England. His talk, On a New General Principle of Analytical Mechanics, was printed in the Report of the meeting published by John Murray, London, in 1843. We give a version of this below.
On a New General Principle of Analytical Mechanics.
By M Jacobi of Königsberg.
In the different problems relative to the motion of a system of material points which have been hitherto considered, one may make an important and curious remark, "that whenever the forces are functions of the coordinates of the moving points only, and the problem is reduced to the integration of a differential equation of the first order of two variables, it may also be reduced to quadratures." Now I have succeeded in proving the general truth of this remark, which appears to constitute a new principle of mechanics. This principle, as well as the other general principles of mechanics, makes known an integral, but with this difference, that whilst the latter give the first integrals of the dynamical differential equations, the new principle gives the last. It possesses a generality very superior to that of other known principles, inasmuch as it applies to cases in which, when the analytical expressions of the forces, as well as the equations by which we express the nature of the system, are composed of the coordinates of the moveables in any manner whatever, principles (such as the principle of the conservation of living forces, of the conservation of areas, and of the conservation, of the centre of gravity) are superior to the new principle in several respects. In, the first place, these principles afford a finite equation between the coordinates of the moveables and the components of their velocities, whilst the integral found by the new principle is simply reduced to quadratures. In the second place, we suppose in the application of the new principle that we have already succeeded in discovering all the integrals but one, a supposition which will be realized in a small number of problems only. This will be sufficient to convince us of the importance of the new principle; but this may be made still more manifest, if I am permitted to illustrate its application by a few examples.
1st. Let us consider the orbit described by a planet in its motion round the sun. The differential equations in dynamical problems being of the second order, we may present them under the form of differential equations of the first order, by introducing the first differentials as new variables. In this manner the determination of the orbit of the planet will depend upon the integration of three differential equations of the first order between four variables. We find two integrals by the principles of living forces (forces vives) and areas. The question is thus reduced to the integration of a single differential equation of two variables and of the first order. Now, by my general theorem, this integration may always be reduced to quadratures. If therefore we choose to reckon this theorem amongst the other principles of mechanics, we see that the general principles of mechanics alone are sufficient to reduce the determination of the orbit of a planet to quadratures.
2nd. Let us consider the motion of a point attracted to two centres of force, after Newton's law of gravitation. The initial velocity being directed in the plane passing through the body and the two centres of attraction, we still have to integrate three differential equations of the first order amongst four variables, one integral of these equations being furnished by the principle of living forces. Euler has discovered another, and thus has succeeded in reducing the problem to a differential equation of the first order between two variables. But this equation was so complicated, that any person but this intrepid geometer would have shrunk from the idea of attempting its integration and reducing it to quadratures. Now, by my general principle, this reduction would have been effected by a general rule without any tentative process, without any extraordinary effort of the mind.
3rd. Let us consider also the famous problem of the rotatory movement of a solid body round a fixed point, the body being under the influence of no accelerating force. In this problem we shall have to integrate five differential equations of the first order amongst six variables. The principle of living forces gives one integral, that of areas gives three others, and the fifth is found by my new principle. We thus see all the integrals of this difficult problem found by the general principles of mechanics alone, without our being required to write a single formula, or even to make a choice of variables.
I will endeavour now to enunciate the rule itself, by means of which the last integration to be effected in the problems of mechanics is found to be reduced to quadratures, the forces being always functions of the coordinates alone. Let us suppose, in the first instance, any system whatsoever of material points entirely free. Let there be found a first integral f' = const., the variables which enter into the function f' being the coordinates of the moveables, and their first differentials taken with respect to the time. I avail myself of the equation
f' = const,for the purpose of eliminating any one of the variables, and I call p' the partial difference of f', taken with respect to this variable. Let f" = const, be a second integral. By means of this equation I eliminate a second variable, and I call p" the partial difference of f" with respect to this variable. Let us suppose that we know all the integrals of the problem but one, and that with respect to each integral f' = const, we seek the corresponding partial difference p with respect to the variable, which we eliminate by means of this integral. The number of variables exceeds by unity that of the integrals: we eliminate by means of each integral a new variable, and we thus succeed in expressing all the variables by means of two of them. Let us call these two variables x and y and x' and y', their first differentials taken with respect to the time. We shall express by means of x and y the quantities x' and y', as well as all the quantities p', p", &c. : since x' and y' are the first differentials of x and y taken with respect to the time, we shall have the equation
y' dx - x' dy = 0,where x' and y' are known functions of the two variables x and y. It is this differential equation, the last of all of them, which we must integrate in order to obtain the complete solution of the problem. Now I show that on dividing this equation by the product of the variables p', p", &c, its first member becomes an exact differential, and therefore the integration of this equation is generally reduced to quadratures.
When we have any system whatsoever of material points, the simplicity of the preceding theorem is in no respect altered, provided we give to the dynamical differential equations that remarkable form under which they have been presented for the first time by the illustrious Astronomer Royal of Dublin, and under which they ought to be presented hereafter in all the general researches of analytical mechanics. It is true that the formulas of Sir W Hamilton are referrible only to the cases where the components of the forces are the partial differences of the same function of the coordinates; but it has not been found to be difficult to make the changes which are necessary in order that these formulas may become applicable to the general case, where the forces are any functions whatever of the coordinates.
When the time enters explicitly into the analytical expressions for the forces, and into the equations of condition of the system, the principle of the final multiplier, found by a general rule, is applicable also to this class of dynamical problems. There are also some particular problems into which enters the resistance of a medium, which give rise to similar theorems. It is the case of a planet revolving round the sun in a medium whose resistance is proportional to any power of the velocity of the planet.
The analysis which has conducted me to the new general principle of analytical mechanics, which I have the honour to communicate to the Association, may be applied to a great number of questions in the integral calculus. I have collected these different applications in a very extensive memoir, which I hope to publish upon my return to Königsberg, and which I shall have the honour of presenting to the Association as soon as it shall be printed.