1. The tendency to apply dynamical principles and methods to explain physical phenomena has steadily increased ever since the discovery of the principle of the Conservation of Energy. This discovery called attention to the ready conversion of the energy of visible motion into such apparently dissimilar things as heat and electric currents, and led almost irresistibly to the conclusion that these too are forms of kinetic energy, though the moving bodies must be infinitesimally small in comparison with the bodies which form the moving pieces of any of the structures or machines with which we are acquainted. As soon as this conception of heat and electricity was reached mathematicians began to apply to them the dynamical method of the Conservation of Energy, and many physical phenomena were shown to be related to each other, and others predicted by the use of this principle; thus, to take an example, the induction of electric currents by a moving magnet was shown by von Helmholtz to be a necessary consequence of the fact that an electric current produces a magnetic field. Of late years things have been carried still further; thus Sir William Thomson in many of his later papers, and especially in his address to the British Association at Montreal on “Steps towards a Kinetic Theory of Matter,” has devoted a good deal of attention to the description of machines capable of producing effects analogous to some physical phenomenon, such, for example, as the rotation of the plane of polarisation of light by quartz and other crystals. For these reasons the view (which we owe to the principle of the Conservation of Energy) that every physical phenomenon admits of a dynamical explanation is one that will hardly be questioned at the present time. We may look on the matter (including, if necessary, the ether) which plays a part in any physical phenomenon as forming a material system and study the dynamics of this system by means of any of the methods which we apply to the ordinary systems in the Dynamics of Rigid Bodies. As we do not know much about the structure of the systems we can only hope to obtain useful results by using methods which do not require an exact knowledge of the mechanism of the system. The method of the Conservation of Energy is such a method, but there are others which hardly require a greater knowledge of the structure of the system and yet are capable of giving us more definite information than that principle when used in the ordinary way. Lagrange's equations and Hamilton's method of Varying Action are methods of this kind, and it is the object of this paper to apply these methods to study the transformations of some of the forms of energy, and to show how useful they are for coordinating results of very different kinds as well as for suggesting new phenomena. A good many of the results which we shall get have been or can be got by the use of the ordinary principle of Thermodynamics, and it is obvious that this principle must have close relations with any method based on considerations about energy. Lagrange’s equations were used with great success by Maxwell in his ‘Treatise on Electricity and Magnetism,’ vol. ii., chaps. 6, 7, 8, to find the equations of the electromagnetic field.
Dynamics and Stability of a Flexible, Slender Cylinder Flexibly Restrained at One End and Free at the Other and Subjected to Axial Flow