On the Second Variation in Certain Anormal Problems of the Calculus of Variations

1941 ◽  
Vol 63 (3) ◽  
pp. 516 ◽  
Author(s):  
E. J. McShane
Keyword(s):  
Calculus Of Variations ◽  
Second Variation ◽  
The Second Variation

scholarly journals Autoparallel Deviation in the Geometry of Lyra

1960 ◽  
Vol 3 (3) ◽  
pp. 263-271 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Calculus Of Variations ◽  
Finsler Manifold ◽  
Geometrical Approach ◽  
Second Variation ◽  
Geodesic Deviation ◽  
The Difference ◽  
The Second Variation

One of the fruitful tools for examining the properties of a Riemannian manifold is the study of “geodesic deviation”. The manner in which a vector, representing the displacement between points on two neighbouring geodesies, behaves gives an indication of the difference between the manifold and an Euclidean space. The study is essentially a geometrical approach to the second variation of the lengthintegral in the calculus of variations [1]. Similar considerations apply in the geometry of Lyra [2] but as we shall see, appropriate analytical modifications must be made. The approach given here is modelled after that of Rund [3] which was originally designed to deal with a Finsler manifold but which applies equally well to the present case.

scholarly journals III. On the discrimination of maxima and minima solutions in the calculus of variations

1886 ◽  
Vol 40 (242-245) ◽  
pp. 476-477
Keyword(s):  
Calculus Of Variations ◽  
Second Variation ◽  
Additional Advantage ◽  
Differential Coefficient ◽  
Undetermined Function ◽  
Independent Variable ◽  
The Second Variation

In the first part of the paper it is shown that the usual investigation by which the second variation of an integral is reduced, requires that the variation given to y (the undetermined function) is such that its differential coefficients, taken with regard to x (the independent variable) are continuous up to the twice- n th order, d n y / dx n being the highest differential coefficient of y appearing in the function to be integrated. But it is not necessary that the variation should be continuous beyond its ( n —1)th differential coefficient, and a method of reducing the variation to Jacobi’s form by a process which is not open to the above objection is then given; and the method has the additional advantage that its simplicity enables it to be easily extended to other cases where there are more than two variables.

scholarly journals Variational formulations of steady rotational equatorial waves

2020 ◽  
Vol 10 (1) ◽  
pp. 534-547
Author(s):  
Jifeng Chu ◽  
Joachim Escher
Keyword(s):  
Linear Stability ◽  
Stream Function ◽  
Water Waves ◽  
Variational Formulation ◽  
Energy Functional ◽  
Equatorial Waves ◽  
Second Variation ◽  
Governing Equations ◽  
The Second Variation ◽  
Variational Formulations

Abstract When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

Integral criteria for closed surfaces with constant mean curvature

2007 ◽  
Vol 463 (2081) ◽  
pp. 1199-1210
Author(s):  
Luca Guzzardi ◽  
Epifanio G Virga
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Second Variation ◽  
Closed Surfaces ◽  
Area Functional ◽  
Symmetric Surface ◽  
The Second Variation ◽  
Integral Identities

We propose three integral criteria that must be satisfied by all closed surfaces with constant mean curvature immersed in the three-dimensional Euclidean space. These criteria are integral identities that follow from requiring the second variation of the area functional to be invariant under rigid displacements. We obtain from them a new proof of the old result by Delaunay, to the effect that the sphere is the only closed axis-symmetric surface.

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