Morse index and bifurcation of p-geodesics on semi Riemannian manifolds

Monica Musso; Jacobo Pejsachowicz; Alessandro Portaluri

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 13, Issue: 3, page 598-621
  • ISSN: 1292-8119

Abstract

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Given a one-parameter family { g λ : λ ∈ [ a , b ] } of semi Riemannian metrics on an n-dimensional manifold M, a family of time-dependent potentials { V λ : λ ∈ [ a , b ] } and a family { σ λ : λ ∈ [ a , b ] } of trajectories connecting two points of the mechanical system defined by ( g λ , V λ ) , we show that there are trajectories bifurcating from the trivial branch σ λ if the generalized Morse indices μ ( σ a ) and μ ( σ b ) are different. If the data are analytic we obtain estimates for the number of bifurcation points on the branch and, in particular, for the number of strictly conjugate points along a trajectory using an explicit computation of the Morse index in the case of locally symmetric spaces and a comparison principle of Morse Schöenberg type.

How to cite

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Musso, Monica, Pejsachowicz, Jacobo, and Portaluri, Alessandro. "Morse index and bifurcation of p-geodesics on semi Riemannian manifolds." ESAIM: Control, Optimisation and Calculus of Variations 13.3 (2007): 598-621. <http://eudml.org/doc/250003>.

@article{Musso2007,
abstract = { Given a one-parameter family $\\{g_\lambda\colon \lambda\in [a,b]\\}$ of semi Riemannian metrics on an n-dimensional manifold M, a family of time-dependent potentials $\\{ V_\lambda\colon \lambda\in [a,b]\\}$ and a family $\\{\sigma_\lambda\colon \lambda\in [a,b]\\} $ of trajectories connecting two points of the mechanical system defined by $(g_\lambda, V_\lambda)$, we show that there are trajectories bifurcating from the trivial branch $\sigma_\lambda$ if the generalized Morse indices $\mu(\sigma_a)$ and $\mu (\sigma_b)$ are different. If the data are analytic we obtain estimates for the number of bifurcation points on the branch and, in particular, for the number of strictly conjugate points along a trajectory using an explicit computation of the Morse index in the case of locally symmetric spaces and a comparison principle of Morse Schöenberg type. },
author = {Musso, Monica, Pejsachowicz, Jacobo, Portaluri, Alessandro},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Generalized Morse index; semi-Riemannian manifolds; perturbed geodesic; bifurcation; generalized Morse index},
language = {eng},
month = {7},
number = {3},
pages = {598-621},
publisher = {EDP Sciences},
title = {Morse index and bifurcation of p-geodesics on semi Riemannian manifolds},
url = {http://eudml.org/doc/250003},
volume = {13},
year = {2007},
}

TY - JOUR
AU - Musso, Monica
AU - Pejsachowicz, Jacobo
AU - Portaluri, Alessandro
TI - Morse index and bifurcation of p-geodesics on semi Riemannian manifolds
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/7//
PB - EDP Sciences
VL - 13
IS - 3
SP - 598
EP - 621
AB - Given a one-parameter family $\{g_\lambda\colon \lambda\in [a,b]\}$ of semi Riemannian metrics on an n-dimensional manifold M, a family of time-dependent potentials $\{ V_\lambda\colon \lambda\in [a,b]\}$ and a family $\{\sigma_\lambda\colon \lambda\in [a,b]\} $ of trajectories connecting two points of the mechanical system defined by $(g_\lambda, V_\lambda)$, we show that there are trajectories bifurcating from the trivial branch $\sigma_\lambda$ if the generalized Morse indices $\mu(\sigma_a)$ and $\mu (\sigma_b)$ are different. If the data are analytic we obtain estimates for the number of bifurcation points on the branch and, in particular, for the number of strictly conjugate points along a trajectory using an explicit computation of the Morse index in the case of locally symmetric spaces and a comparison principle of Morse Schöenberg type.
LA - eng
KW - Generalized Morse index; semi-Riemannian manifolds; perturbed geodesic; bifurcation; generalized Morse index
UR - http://eudml.org/doc/250003
ER -

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