5i6 



NATURE 



[September 29, 1892 



refractive index of 1-53. The equiangular prisms cause 

 less loss of light by absorption and reflection than either 

 the spherical or Fresnel refractors, and also act on the 

 light so that ex-focal light is better dealt with, thereby 

 reducing the divergence. 



Charles A. Stevenson. 



MODERN DYNAMICAL METHODS. 



A DYNAMICAL system is said to possess a given 

 ■^~*- number of degrees of freedom, when it is capable of 

 assuming the same number of independent positions. 

 The position of the system, in any possible configuration, 

 is capable of being determined by a definite number of 

 independent quantities, which are equal to the number of 

 degrees of freedom of the system. These quantities are 

 called the co-ordinates of the system. 



\Vhen the system possesses six degrees of freedom, the 

 motion may be completely determined by expressing in 

 mathematical language a principle which may be con- 

 veniently termed \.\\& principle of momentum. This prin- 

 ciple is specified by the following two propositions : — 

 (i.) The rate of change of the component of the linear 

 momentum, parallel to an axis, of any dynamical system, 

 is equal to the component, parallel to that axis, of the im- 

 pressed forces which act upon the system ; (ii.) the rate of 

 change of the component of the angular 7nomentum about 

 any axis, is equal to the moment of the impressed forces 

 about that axis. Since the motion pf the system may be 

 referred to any set of fixed or moving rectangular axes, 

 the above-mentioned dynamical principle furnishes six 

 equations connecting the six co-ordinates, which, when 

 integrated, will determine the latter in terms of the time 

 and the initial circumstances of the motion. 



The various ways of expressing this dynamical prin- 

 ciple in mathematical language are explained in treatises 

 on dynamics : and a variety of special forms and par- 

 ticular cases are obtained, by means of which the solu- 

 tion of numerous problems can be simplified. For 

 example, Euler's equations, for determining the motion 

 of rotation of a single rigid body about its centre of 

 inertia, is a particular case of the second proposition ; 

 whilst Kirchhoff's equations, for determining the motion 

 of a single solid in an infinite liquid, is a special form of 

 both propositions. 



When a conservative system possesses seven degrees 

 of freedom, the motion may be completely determined by 

 means of the principle of momentum combined with the 

 principle of energy. The first principle, as we have 

 already shown, furnishes six equations, whilst the second 

 furnishes one ; hence, we have a sufficient number of 

 equations for determining the motion. 



When a dynamical system possesses more than seven 

 degrees of freedom, the principles of momentum and 

 energy are insufficient to determine the motion ; and 

 under these circumstances, the most convenient method 

 to adopt is to use Lagrange's equations ; but inasmuch 

 as these equations are double-edged tools, which are apt 

 to cut the fingers of the unwary, their employment re- 

 quires considerable care. 



The kinetic energy of a dynamical system can be ex- 

 pressed in a variety of different forms, but it will only be 

 necessary to mention the following three. In the first 

 form, it is expressed as a homogeneous quadratic function 

 of velocities, which are the time-variations of the co- 

 ordinates of the system. This form, which will be de- 

 noted by T, is called the Lagrangian form ; it is the 

 only one which it is permissible to use when employing 

 Lagrange's equations, and many mistakes have been 

 made by persons who have attempted to use some other 

 form. 



In the second form, which is called the Hamiltonian 

 form, the kinetic energy is expressed as a homogeneous 



NO. 1 1 96, VOL. 46] 



quadratic function of the momenta of the system. If 

 6 be any co-ordinate, and the generalized momentum 

 of type 6, it is known that 



S-e (■) 



whence e is a linear function of the velocities. Hence,, 

 if the velocities be eliminated from the Lagrangian ex- 

 pression for the kinetic energy by means of (i), it follows 

 that the latter will be expressible as a homogeneous 

 quadratic function of the momenta G, which is the 

 Hamiltonian form. We shall denote this form by C^. 

 Lagrange's equations are 



dj^T\ _ 9T _ _ ay , . 



where V is the potential energy ; and if the elimination 

 be performed, we shall obtain 



de.dZ ^ _ dV 



dt "•" de dd 



(3) 



(4) 



(5) 



we have also the reciprocal relation 



'^^e 



Equations (3) and (4) are Hamilton's equations of 

 motion. 



The third form of the expression for the kinetic energy 

 is of special iinporlance in hydrodynamics and other 

 branches of physics. It sometimes happens that a 

 quantity occurs which can be recognized as a momentum, 

 or as a quantity in the nature of a momentum, whilst the 

 velocity corresponding to this momentum is either un- 

 known or would be inconvenient to introduce. This 

 occurs in problems relating to the motion of perforated 

 solids in a liquid, when there is circulation, and is a par- 

 ticular case of Dr. Routh's theory of the " Ignoration of 

 Velocities." ^ We therefore require a form of Lagrange's 

 equations in which certain velocities are eliminated, and 

 are replaced by the corresponding momenta. 



Let the co-ordinates of the system be divided into two 

 groups, 6 and x 5 ^^^ ^^^ '^ denote the generalized mo- 

 mentum corresponding to x- Then 

 dT 



BJ=' 



By means of (5) all the velocities x can be eliminated 

 from the expression for the kinetic energy ; and it is re- 

 markable, that the result of the elimination does not con- 

 tain any products of the form k6. The expression for T 

 may accordingly be written 



T=2; + S (6) 



where X is a homogeneous quadratic function of the 

 velocities 6, and ^ is a similar function of the mo- 

 menta K. 



Equation (6) is therefore a mixed form, which is partly 

 Lagrangian and partly Hamiltonian. We now require 

 the corresponding form of the equations of motion in 

 which all the x's have been eliminated from Lagrange's 

 equations. 



From (i) it follows that the generalized momentum 

 is a linear function of the velocities B, x ; and if the latter 

 velocities be eliminated by means of (5), it follows that 

 is expressible as a linear function of 6, k. Let the 

 portion which is a linear function of the k's be denoted by 

 ; then it can be shown, that if 



L = 2; + 2(0^)-^- V .... (7) 

 the equation of type is 



ddL dL _ 



dtdd dd 



(8) 



I Having regard to the object of the theory, I think the phrase " Ig- 

 noration of Velocities " is better than " Ignoration of Co-ordinates." 



