Singularities of caustics and Sturm theory

1994 ◽  
pp. 39-46
Author(s):  
V. Arnold
Keyword(s):  
Sturm Theory

scholarly journals Theorem on six vertices of a plane curve via Sturm theory

1997 ◽  
pp. 257-266 ◽  
Author(s):  
L. Guieu ◽  
E. Mourre ◽  
V. Yu Ovsienko
Keyword(s):  
Plane Curve ◽  
Sturm Theory

scholarly journals Some sturm theory

1981 ◽  
Vol 39 (1) ◽  
pp. 296-296
Author(s):  
Walter Leighton
Keyword(s):  
Sturm Theory

Multiparameter Sturm theory

1984 ◽  
Vol 99 (1-2) ◽  
pp. 173-184 ◽  
Author(s):  
Paul Binding ◽  
Patrick J. Browne
Keyword(s):  
Elliptic Partial Differential Equation ◽  
Comparison Theorems ◽  
Unified Theory ◽  
Focal Points ◽  
The Matrix ◽  
Sturm Liouville ◽  
Left And Right ◽  
Image Position ◽  
Completeness Of Eigenfunctions ◽  
Sturm Theory

SynopsisLet Sturm–Liouville problemswith continuous coefficients and appropriate boundary conditions, be coupled by the eigenvalue λ = (λ1, … λk). When k = 1, there are various oscillation, perturbation and comparison theorems concerning existence and continuous or monotonic dependence of eigenvalues, eigenfunctions and their zeros (i.e. focal points).We attempt a unified theory for such results, valid for general fc, under conditions known as "left" and “right” definiteness. A representative result may be stated loosely as follows: if LD holds then (elementwise) monotonic dependence of p, q and the matrix [ars] forces monotonic dependence of λ. LD is a generalisation of the “polar” case for k = 1, and was originally conceived for a quite different purpose, viz. completeness of eigenfunctions via elliptic partial differential equation theory.

scholarly journals Sturm–Liouville problems with eigenparameter dependent boundary conditions

1994 ◽  
Vol 37 (1) ◽  
pp. 57-72 ◽  
Author(s):  
P. A. Binding ◽  
P. J. Browne ◽  
K. Seddighi
Keyword(s):  
Boundary Conditions ◽  
Liouville Problem ◽  
Asymptotic Estimates ◽  
Comparison Results ◽  
Sturm Liouville ◽  
Image Position ◽  
Sturm Theory

Sturm theory is extended to the equationfor 1/p, q, r∈L1 [0, 1] with p, r > 0, subject to boundary conditionsandOscillation and comparison results are given, and asymptotic estimates are developed. Interlacing of eigenvalues with those of a standard Sturm–Liouville problem where the boundary conditions are ajy(j) = cj(py′)(j), j=0, 1, forms a key tool.

scholarly journals Sturm theory with applications in geometry and classical mechanics

2021 ◽  
Author(s):  
Vivina L. Barutello ◽  
Daniel Offin ◽  
Alessandro Portaluri ◽  
Li Wu
Keyword(s):  
Phase Plane ◽  
Classical Mechanics ◽  
Maslov Index ◽  
Comparison Theorems ◽  
Geometrical Properties ◽  
Lagrangian Subspace ◽  
Higher Dimensional ◽  
Lagrangian Subspaces ◽  
Geometrical Information ◽  
Sturm Theory

AbstractClassical Sturm non-oscillation and comparison theorems as well as the Sturm theorem on zeros for solutions of second order differential equations have a natural symplectic version, since they describe the rotation of a line in the phase plane of the equation. In the higher dimensional symplectic version of these theorems, lines are replaced by Lagrangian subspaces and intersections with a given line are replaced by non-transversality instants with a distinguished Lagrangian subspace. Thus the symplectic Sturm theorems describe some properties of the Maslov index. Starting from the celebrated paper of Arnol’d on symplectic Sturm theory for optical Hamiltonians, we provide a generalization of his results to general Hamiltonians. We finally apply these results for detecting some geometrical information about the distribution of conjugate and focal points on semi-Riemannian manifolds and for studying the geometrical properties of the solutions space of singular Lagrangian systems arising in Celestial Mechanics.

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