TY - JOUR

T1 - Sturm theory with applications in geometry and classical mechanics

AU - Barutello, Vivina L.

AU - Offin, Daniel

AU - Portaluri, Alessandro

AU - Wu, Li

N1 - Publisher Copyright:
© 2021, The Author(s).

PY - 2021/10

Y1 - 2021/10

N2 - Classical 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.

AB - Classical 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.

KW - Conjugate points

KW - Conley-Zehnder index

KW - Hamiltonian dynamics

KW - Kepler problem

KW - Maslov index

UR - http://www.scopus.com/inward/record.url?scp=85099972465&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85099972465&partnerID=8YFLogxK

U2 - 10.1007/s00209-020-02686-3

DO - 10.1007/s00209-020-02686-3

M3 - Article

AN - SCOPUS:85099972465

SN - 0025-5874

VL - 299

SP - 257

EP - 297

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 1-2

ER -