A History of the Progress of the Calculus of Variations During the Nineteenth Century |
From inside the book
Results 1-5 of 100
Page 1
... ordinary maxima and minima problems in the preceding pages of his work , Lagrange states that the same principles may be applied to determine curves which possess at every point some assigned maximum or minimum property . For example ...
... ordinary maxima and minima problems in the preceding pages of his work , Lagrange states that the same principles may be applied to determine curves which possess at every point some assigned maximum or minimum property . For example ...
Page 2
... ordinary principles of maxima and minima problems we differentiate the proposed expression with respect to y ' as variable , and equate the differential coefficient to zero . This gives ( m − x ) { y + ( n − x ) y ' } + ( n − x ) { y ...
... ordinary principles of maxima and minima problems we differentiate the proposed expression with respect to y ' as variable , and equate the differential coefficient to zero . This gives ( m − x ) { y + ( n − x ) y ' } + ( n − x ) { y ...
Page 3
... ordinary notation instead of Lagrange's . Let p denote dy and dx ' suppose f ( x , y , p ) to represent any function the integral of which taken between certain fixed limits is to have a maximum or minimum value . Change y into y + dy ...
... ordinary notation instead of Lagrange's . Let p denote dy and dx ' suppose f ( x , y , p ) to represent any function the integral of which taken between certain fixed limits is to have a maximum or minimum value . Change y into y + dy ...
Page 8
... ordinary conditions for the maximum or minimum value of fVdx , where V is supposed to con- tain x and y , and the differential coefficients of y with respect to x . In his investigation he first supposes that x itself does not receive ...
... ordinary conditions for the maximum or minimum value of fVdx , where V is supposed to con- tain x and y , and the differential coefficients of y with respect to x . In his investigation he first supposes that x itself does not receive ...
Page 9
... ordinary notation instead of Lagrange's . Suppose V a function of x , y , z , p , q , r , s , = dz Q dz t , ... where pdx ' = dy ' let U = ffVdyda ; then r = d2z dx2 " SU = ff & Vdydx , dV dV dV and SV = бz + dz dp say 8 = d2z dx dy ...
... ordinary notation instead of Lagrange's . Suppose V a function of x , y , z , p , q , r , s , = dz Q dz t , ... where pdx ' = dy ' let U = ffVdyda ; then r = d2z dx2 " SU = ff & Vdydx , dV dV dV and SV = бz + dz dp say 8 = d2z dx dy ...
Other editions - View all
Common terms and phrases
arbitrary constants axis Calculus of Variations catenary chapter co-ordinates condition considered contains curvature curve Delaunay denote determined Differential Calculus differential equation double integral ds ds dv dv dx dx dx dy dz dx dz dx² dy dx dy dy dy dz dx dz dy dz dz Euler exact differential coefficient example expression formula geodesic gives Hence indefinitely small independent variable Integral Calculus integral sign investigation involves Jacobi's Jacobi's theorem Lacroix Lagrange maxima and minima maximum or minimum memoir method minimum area minimum value notation obtain occupies pages occur Ostrogradsky partial differential equation plane Poisson problem quantities remarks respect result Sarrus second order shew solution Stegmann Strauch suppose surface theorem tion treatise triple integral vanish volume Y₁ zero
Popular passages
Page 140 - It is true that one of the fundamental principles of this method consists in removing as much as possible the differential coefficients of the variations which occur under the integral sign; but the calculus of variations only indicates this operation and refers the execution of it to the Integral Calculus. CHAPTER VI. DELAUNAY. 133. THE Academy of Sciences at Paris proposed the following as the subject of competition for their great mathematical prize in 1842 ; To find the limiting equations which...
Page 229 - Jacobi in 1837, and the memoir which Jacobi then published has given rise to an extensive series of commentaries and developments. Before however we proceed to Jacobi's investigations, we will give an analysis of Legendre's memoir and of some others connected with it. • 197. Legendre's memoir is entitled Memoire sur la maniere de distinguer les maxima des minima dans le Calcul des Variations.
Page 142 - ... can be calculated. It will be necessary to effect successive integrations, and to take each integral between appropriate limits, and these can be determined in the following manner. The order of the successive integrations being arbitrary, we can suppose that we integrate first with respect to z, then with respect to y, and then with respect to x. In the first integration y and x are regarded as constants, and the integration with respect to z extends over all the values of z which render f(x,...
Page 37 - ... of Variations involving the variation of a certain double integral, the limits of the integration being also variable; it is the earliest example of the solution of such a problem. Gauss himself says on page 67, " Sed quum calculus variationum integralium duplicium pro casu ubi etiam limites tanquam variabiles spectari debent, hactenus parum excultus sit, hanc disquisitionem subtilem paullo profundius petere oportet.
Page 345 - Bjorling discusses another particular example before considering the general equation, namely, among all surfaces which can be formed by the motion of a straight line which always remains parallel to a fixed plane, to determine that of minimum area.
Page 286 - Let there be any linear differential expression which involves x, y, and the differential coefficients of y with respect to x...
Page 409 - ... that y is then a minimum and so is F. These results do not agree with those in the book. The case in which y = k seems there overlooked. If y = k we have h = k = a. And it may be seen that the relation on the 14th line of page 165 of the book may be satisfied by supposing a = y and the angle CPY zero. 352. On page 365 the following problem is suggested ; to construct upon a given base a curve such that the superficial area of the surface generated by its revolution round AB may be given, and...