Page:EB1911 - Volume 27.djvu/943

This discussion yields an important result, which may be stated as follows: Let two stationary curves of the integral be drawn from the same initial point A to points P, Q, which are near together, and let the line PQ be of length ν, and make an angle ω with the axis of x (fig. 1). The excess of the integral taken along AQ, from A to Q, above the integral taken along AP, from A to P, is expressed, correctly to the first order in ν, by the formula

$$\nu \cos \omega \left\{F(x,y,y')+(\tan \omega -y'){\frac {\partial F}{\partial y'}}\right\}$$

Fig. 1.

Fig. 1.

In this formula x, y are the co-ordinates of P, and y ′ has the value belonging to the point P and the stationary curve AP. When the coefficient of ν cos ω in the formula vanishes, the curve AP is said to be cut transversely by the line PQ, and a curve which cuts a family of stationer curves transversely is described as a transversal of those curves. In the problem of variable limits, when a terminal point moves on a given guiding curve, the integral cannot be an extremum unless the stationary curve along which it is taken is cut transversely by the guiding curve at the terminal point. A simple example is afforded by the shortest line, drawn on a surface, from a point to a given curve, lying on the surface. The required curve must be a geodesic, and it must cut the given curve at right angles.

The problem of variable limits may always be treated by a method of which the following is the principle: in the First Problem let the initial point (x₀, y₀) be fixed, and let the terminal point (x₁, y₁) move on a fixed guiding curve C₁. Now, whatever method the terminal point may be, the integral cannot be an extremum unless it is taken along a stationary curve. We have then to choose among those stationary curves which are drawn from (x₀, y₀) to points of C₁ that one which makes the integral an extremum. This can be done by expressing the value of the integral taken along a stationary curve from the point (x₀, y₀) to the point (x₁, y₁) in terms of the co-ordinates x₁, y₁, and then making this expression an extremum, in regard to variations ofx₁, y₁, by the methods of the differential calculus, subjecting (x₁, y₁) to the condition of moving on the curve C₁.

An important example of the first variation of integrals is afforded by the Principle of Least Action in dynamics. The kinetic energy T is a homogeneous function of the second degree in the differential coefficients q₁, q₂, . . . q_n of the co-ordinates q₁, q₂, . . . q_n with respect to the time t, and the potential energy V is a function of these co-ordinates. The energy equation is of the form Principle

T+V＝E,

where E is a constant. A course of the system is defined when the coordinates q are expressed as functions of a single parameter θ. The action A of the system is defined as the integral ∫t₁/t₀Tdt, taken along a course from the initial position (q⁽⁰⁾) to the final position (q⁽⁰⁾), but t₀ and t₁ are not fixed. The equations of motion are the principal equations answering to this integral. To obtain them it is most convenient to write Φ(q) for T, and to express the integral in the form

$\int _{\theta _{0}}^{\theta _{1}}2(E-V){\tfrac {1}{2}}\{\Phi (q')\}{\tfrac {1}{2}}d\theta$

where q′ denotes the differential coefficient of a co-ordinate q with respect to θ, and, in accordance with the parametric method, the limits of integration are fixed, and the integrand is a homogeneous function of the q′’s of the first degree. There is then no difficulty in deducing the Lagrangian equations of motion of the type

$${\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q}}}}-{\frac {\partial T}{\partial q}}+{\frac {\partial V}{\partial q}}=0.$$ These equations determine the actual course of the system. Now if the system, in its actual course, passes from a given initial position (q⁽⁰⁾) to a variable final position (q), the action A becomes a function of the q’s, and the first method used in the problem of variable limits shows that, for every q

$${\frac {\partial A}{\partial q}}={\frac {\partial T}{\partial {\dot {q}}}}.$$

When the kinetic energy T is expressed as a homogeneous quadratic function of the momenta $\partial T/\partial {\dot {q}}$, say

T＝1/2Σ_r, . . .

and the differential coefficients of A are introduced instead of those of T, the energy equation becomes a non-linear partial differential equation of the first order for the determination of A as a function of the q’s. This equation is Principle of varying action.

1/2Σ_r, . . .

A complete integral of this equation would yield an expression for A as a function of the q's containing n arbitrary constants, al, al, . a., of which one a, . is merely additive to A; and the courses of the system which are compatible with the equations of motion are determined by equations of the form

∂A/∂a₁＝...

where the b’s are new arbitrary constants. It is noteworthy that the differential equations of the second order by which the geodesics on an ellipsoid are determined were first solved by this method (C. G. ]. Jacobi, 1839).

It has been proved that every problem of the calculus of variations, in which the integral to be made an extremum contains only one independent variable, admits of a similar transformation; that is to say, the integrals of the principal p"]”dp'° equations can always be obtained, in the way described “f “W” above, from a complete integral of a partial differential mei equation of the first order, and this partial differential 8 'mal e uation can always be formed by a process of elimination. 'fegjr Tiiiese results were first proved by A. Clebsch (1858).

Among other analytical developments of the theory of the first variation we may note that the necessary and sufficient condition that an expression of the form A

F(x. y, y', - - -y<"') ffzjfgf"

should be the differential coefficient of another expression, of the form

F1(x1 yr y,1- ' ~y<n '1))

is the identical vanishing of the expression ∂F d 6F dz 6F d" ∂F

3-Z6 5;/+52 gr- ~ - +(~I)" gngwvfy

The result was first found by Euler (1744). A differential equation

¢<x, yy yr, yy):O gjlxdifkll

is the principal equation answering to an integral of the form, ential

fF(x, yu y)dx equation

if the equation may gl-is¢

ri Q§ ; i(} from a

ax ay” 'ay' problem

is satisfied identically. In the more general case of an °f'“'7 equation of the form °' C” "S

~ of varia¢(x,

y, y', . . .y(2 )) = O tions.

the corresponding condition is that the' differential expression obtained by Lagrange's process of variation, viz., gi aa lg a¢ Way

ay 'WJVW dx +' ' ' 'l'ay<2'~> dx2"

Y

must be identical with the “ adjoint ' differential expression it '£<§ ;¢:) § <@ 'il< 3¢

ay 5y"'dx ayI5fY 'l'dxz ay” 5fY)'»~~'l'dx2» 63/(2n)6y-This matter has been very fully investigated by A. Hirsch (1897). To illustrate the transformation of the first variation of multiple integrals we consider a double integral of the form D3/>(x, y, 2, P. q. 1, S, ilfixfly.

taken over that area of the z plane which is bounded by a Tggguon closed curve s'. Here p, q, t denote the partial of a differential coefficients of z with respect to x and y of doume the first and second orders, according to the usual notation. When z is changed into z-1-ew, the terms of the first order in e are

GW 6// Bw 61,0 ow of 62w 6l/ 6271! if 327.0 FU (asm ap ax+& ay'l'af ae +53 axay+ an ay2 dxdy Each term must be transformed so that no differential coefficients of w are left under the sign of double integration. We exemplify the process by taking the term containing 6210/6x2. We have "9¢""w iiixfif) 3 <Q'£>.i'i

K/5? @dxdy'./flbx <6r 3x Tax 61' éx dxdy 3 dxl/ dw 3 6 3(/ 62 <6.b>']

f/'LE <52 ax) ' ax i w5¢ (af) i 'l'wax2. af dxdy

The first two terms are transformed into a line integral taken round the boundary s', and we thus find as

where ν denotes the direction of the normal to the edge s' drawn outwards. The double integral on the right-hand side contributes a term to the principal equation, and the line integral contributes terms to the boundary conditions. The line integral admits of further transformation by' means of the relations a a

§ =%' cos (xl v)-5373 COS fy. V).

/cos(x, v)cos(y, 'v)% $5 ds'= cos(x, v)cos(y, 10% wd$'.