{"id":{"repo_id":"aachen","oai_identifier":"oai:publications.rwth-aachen.de:59205"},"canonical_url":"https://search.dev.ndltd.org/etd/aachen/oai:publications.rwth-aachen.de:59205","repository":{"repo_id":"aachen","name":"RWTH Aachen University","base_url":"https://publications.rwth-aachen.de/oai2d"},"display":{"title":"A KAM theorem for the spatial lunar problem","abstract":"The spatial lunar problem describes the motion of a small moon in three dimensional space close to its planet. It is influenced by the gravitational attraction of the planet and the sun. The Hamiltonian function is a small perturbation of the maximally superintegrable Kepler Hamiltonian K. To analyze the dynamics of the given three-degree-of-freedom system we normalize it, that is we approximate it by a simplified system. Using Kolmogorov-Arnol'd-Moser (KAM) Theory we show that the invariant 3-tori of the normalized system persist in the original lunar problem. It is a well known fact, that the lunar problem can be normalized twice, such that the twice normalized and truncated Hamiltonian is a power series of the perturbation parameter ε and has two first integrals which correspond to the third component of the angular momentum and to the Kepler Hamiltonian K. Alike holds for the \"orbiting dust problem\" and the \"hydrogen atom in crossed fields\". The resulting normalized and truncated Hamiltonians of all three problems differ merely in some coefficients. So, we focus on the lunar problem as an example for a certain \"class\" of perturbed Keplerian systems. Using the fact, that we have obtained a Liouville integrable system by normalizing and truncating, we construct action-angle variables (J, φ). The classical KAM theorem applies to non-degenerate n-degree-of-freedom systems: If the Hamiltonian has the form H[sub]0[/sub]+εH[sub]1[/sub]+O(ε[sup]2[/sup]) then the determinant of the Hessian of the unperturbed part H[sub]0[/sub] with respect to the action variables must not vanish. This yields that the so called frequency map ist a local diffeomorphism and all those n-tori whose frequencies satisfy a certain Diophantine condition survive small perturbations. The persisting n-tori form a set whose complement has measure O(α).The Hamiltonian H= K+ε H[sub]1[/sub]+ε[sup]2[/sup] H[sub]2[/sub]+hot of the spatial lunar problem is (maximally) superintegrable or properly degenerate. Its unperturbed part, the Kepler Hamiltonian K, depends only on one of three action variables. The classical KAM theorem does not apply. A properly degenerate Hamiltonian H= H[sub]0[/sub]+ε H[sub]1[/sub]+O(&epsilon,[sup]2[/sup]) has a perturbation that removes the degeneracy, if the unperturbed part H[sub]0[/sub] depends on r [ n action variables, is non-degenerate with respect to those and the first order term of the perturbation H[sub]1[/sub] depends non-degenerately on the remaining n-r action variables. For such systems the frequency map of the intermediate system with Hamiltonian H[sub]0[/sub]+ε H[sub]1[/sub] is a diffeomorphism, while the frequencies that occur have two orders of magnitude. For such systems a KAM theorem similar to the classical one exists. The Diophantine condition has to be changed in such a way that it depends linearly on ε. In the case of the lunar problem, the perturbation of the Kepler Hamiltonian does not remove the degeneracy. We have to add the second order term of the perturbation H[sub]2[/sub] to the intermediate Hamiltonian to obtain a dependence on all three action variables. No known KAM theorem applies. But the Kepler Hamiltonian depends only on one action variable and it does so linearly. Using this fact, we derive and prove a new KAM theorem for our class of Hamiltonian systems by adapting the way this has been done for proper degenerate systems whose perturbations remove the degeneracy. Applying this new theorem to the lunar problem yields the persistence of most of the invariant 3-tori of the twice normalized, truncated system in the original one.","abstract_html":"The spatial lunar problem describes the motion of a small moon in three dimensional space close to its planet. It is influenced by the gravitational attraction of the planet and the sun. The Hamiltonian function is a small perturbation of the maximally superintegrable Kepler Hamiltonian K. To analyze the dynamics of the given three-degree-of-freedom system we normalize it, that is we approximate it by a simplified system. Using Kolmogorov-Arnol'd-Moser (KAM) Theory we show that the invariant 3-tori of the normalized system persist in the original lunar problem. It is a well known fact, that the lunar problem can be normalized twice, such that the twice normalized and truncated Hamiltonian is a power series of the perturbation parameter ε and has two first integrals which correspond to the third component of the angular momentum and to the Kepler Hamiltonian K. Alike holds for the "orbiting dust problem" and the "hydrogen atom in crossed fields". The resulting normalized and truncated Hamiltonians of all three problems differ merely in some coefficients. So, we focus on the lunar problem as an example for a certain "class" of perturbed Keplerian systems. Using the fact, that we have obtained a Liouville integrable system by normalizing and truncating, we construct action-angle variables (J, φ). The classical KAM theorem applies to non-degenerate n-degree-of-freedom systems: If the Hamiltonian has the form H[sub]0[/sub]+εH[sub]1[/sub]+O(ε[sup]2[/sup]) then the determinant of the Hessian of the unperturbed part H[sub]0[/sub] with respect to the action variables must not vanish. This yields that the so called frequency map ist a local diffeomorphism and all those n-tori whose frequencies satisfy a certain Diophantine condition survive small perturbations. The persisting n-tori form a set whose complement has measure O(α).The Hamiltonian H= K+ε H[sub]1[/sub]+ε[sup]2[/sup] H[sub]2[/sub]+hot of the spatial lunar problem is (maximally) superintegrable or properly degenerate. Its unperturbed part, the Kepler Hamiltonian K, depends only on one of three action variables. The classical KAM theorem does not apply. A properly degenerate Hamiltonian H= H[sub]0[/sub]+ε H[sub]1[/sub]+O(&epsilon,[sup]2[/sup]) has a perturbation that removes the degeneracy, if the unperturbed part H[sub]0[/sub] depends on r [ n action variables, is non-degenerate with respect to those and the first order term of the perturbation H[sub]1[/sub] depends non-degenerately on the remaining n-r action variables. For such systems the frequency map of the intermediate system with Hamiltonian H[sub]0[/sub]+ε H[sub]1[/sub] is a diffeomorphism, while the frequencies that occur have two orders of magnitude. For such systems a KAM theorem similar to the classical one exists. The Diophantine condition has to be changed in such a way that it depends linearly on ε. In the case of the lunar problem, the perturbation of the Kepler Hamiltonian does not remove the degeneracy. We have to add the second order term of the perturbation H[sub]2[/sub] to the intermediate Hamiltonian to obtain a dependence on all three action variables. No known KAM theorem applies. But the Kepler Hamiltonian depends only on one action variable and it does so linearly. Using this fact, we derive and prove a new KAM theorem for our class of Hamiltonian systems by adapting the way this has been done for proper degenerate systems whose perturbations remove the degeneracy. Applying this new theorem to the lunar problem yields the persistence of most of the invariant 3-tori of the twice normalized, truncated system in the original one.","abstract_has_math":false,"creators":["Sommer, Britta Sylvia"],"institution":"Publikationsserver der RWTH Aachen University","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Enß, Volker"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2003,"date_issued":"2003","date_published":"2003","updated_at":"2026-07-30T19:42:39Z","subjects":["info:eu-repo/classification/ddc/530","Dreikörperproblem","KAM-Theorie","Physik","Dynamische Systeme","Hamiltonsche Systeme","KAM Theorie","Störungstheorie"],"languages":["eng"],"rights":["info:eu-repo/semantics/openAccess"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-121012%22"],"render_values":[{"text":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-121012%22","href":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-121012%22","code":true}]}]},"links":{"outbound_url":"https://publications.rwth-aachen.de/record/59205","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Enß, Volker"]},{"key":"dc:creator","label":"Author","values":["Sommer, Britta Sylvia"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:coverage","label":"Dc Coverage","values":["DE"]},{"key":"dc:date","label":"Dc Date","values":["2003"]},{"key":"dc:publisher","label":"Institution","values":["Publikationsserver der RWTH Aachen University"]},{"key":"dc:relation","label":"Dc Relation","values":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-7357","info:eu-repo/semantics/altIdentifier/doi/10.18154/RWTH-CONV-121012"]},{"key":"dc:type","label":"Dc Type","values":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["info:eu-repo/classification/ddc/530","Dreikörperproblem","KAM-Theorie","Physik","Dynamische Systeme","Hamiltonsche Systeme","KAM Theorie","Störungstheorie"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["info:eu-repo/semantics/openAccess"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/record/59205","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-121012%22"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The spatial lunar problem describes the motion of a small moon in three dimensional space close to its planet. It is influenced by the gravitational attraction of the planet and the sun. The Hamiltonian function is a small perturbation of the maximally superintegrable Kepler Hamiltonian K. To analyze the dynamics of the given three-degree-of-freedom system we normalize it, that is we approximate it by a simplified system. Using Kolmogorov-Arnol'd-Moser (KAM) Theory we show that the invariant 3-tori of the normalized system persist in the original lunar problem. It is a well known fact, that the lunar problem can be normalized twice, such that the twice normalized and truncated Hamiltonian is a power series of the perturbation parameter ε and has two first integrals which correspond to the third component of the angular momentum and to the Kepler Hamiltonian K. Alike holds for the \"orbiting dust problem\" and the \"hydrogen atom in crossed fields\". The resulting normalized and truncated Hamiltonians of all three problems differ merely in some coefficients. So, we focus on the lunar problem as an example for a certain \"class\" of perturbed Keplerian systems. Using the fact, that we have obtained a Liouville integrable system by normalizing and truncating, we construct action-angle variables (J, φ). The classical KAM theorem applies to non-degenerate n-degree-of-freedom systems: If the Hamiltonian has the form H[sub]0[/sub]+εH[sub]1[/sub]+O(ε[sup]2[/sup]) then the determinant of the Hessian of the unperturbed part H[sub]0[/sub] with respect to the action variables must not vanish. This yields that the so called frequency map ist a local diffeomorphism and all those n-tori whose frequencies satisfy a certain Diophantine condition survive small perturbations. The persisting n-tori form a set whose complement has measure O(α).The Hamiltonian H= K+ε H[sub]1[/sub]+ε[sup]2[/sup] H[sub]2[/sub]+hot of the spatial lunar problem is (maximally) superintegrable or properly degenerate. Its unperturbed part, the Kepler Hamiltonian K, depends only on one of three action variables. The classical KAM theorem does not apply. A properly degenerate Hamiltonian H= H[sub]0[/sub]+ε H[sub]1[/sub]+O(&epsilon,[sup]2[/sup]) has a perturbation that removes the degeneracy, if the unperturbed part H[sub]0[/sub] depends on r [ n action variables, is non-degenerate with respect to those and the first order term of the perturbation H[sub]1[/sub] depends non-degenerately on the remaining n-r action variables. For such systems the frequency map of the intermediate system with Hamiltonian H[sub]0[/sub]+ε H[sub]1[/sub] is a diffeomorphism, while the frequencies that occur have two orders of magnitude. For such systems a KAM theorem similar to the classical one exists. The Diophantine condition has to be changed in such a way that it depends linearly on ε. In the case of the lunar problem, the perturbation of the Kepler Hamiltonian does not remove the degeneracy. We have to add the second order term of the perturbation H[sub]2[/sub] to the intermediate Hamiltonian to obtain a dependence on all three action variables. No known KAM theorem applies. But the Kepler Hamiltonian depends only on one action variable and it does so linearly. Using this fact, we derive and prove a new KAM theorem for our class of Hamiltonian systems by adapting the way this has been done for proper degenerate systems whose perturbations remove the degeneracy. Applying this new theorem to the lunar problem yields the persistence of most of the invariant 3-tori of the twice normalized, truncated system in the original one."]},{"key":"dc:source","label":"Dc Source","values":["Aachen : Publikationsserver der RWTH Aachen University 147 S. (2003). doi:10.18154/RWTH-CONV-121012 = Aachen, Techn. Hochsch., Diss., 2003"]},{"key":"dc:title","label":"Title","values":["A KAM theorem for the spatial lunar problem"]}]}],"canonical_facts":{"dc:contributor":["Enß, Volker"],"dc:coverage":["DE"],"dc:creator":["Sommer, Britta Sylvia"],"dc:date":["2003"],"dc:description":["The spatial lunar problem describes the motion of a small moon in three dimensional space close to its planet. It is influenced by the gravitational attraction of the planet and the sun. The Hamiltonian function is a small perturbation of the maximally superintegrable Kepler Hamiltonian K. To analyze the dynamics of the given three-degree-of-freedom system we normalize it, that is we approximate it by a simplified system. Using Kolmogorov-Arnol'd-Moser (KAM) Theory we show that the invariant 3-tori of the normalized system persist in the original lunar problem. It is a well known fact, that the lunar problem can be normalized twice, such that the twice normalized and truncated Hamiltonian is a power series of the perturbation parameter ε and has two first integrals which correspond to the third component of the angular momentum and to the Kepler Hamiltonian K. Alike holds for the \"orbiting dust problem\" and the \"hydrogen atom in crossed fields\". The resulting normalized and truncated Hamiltonians of all three problems differ merely in some coefficients. So, we focus on the lunar problem as an example for a certain \"class\" of perturbed Keplerian systems. Using the fact, that we have obtained a Liouville integrable system by normalizing and truncating, we construct action-angle variables (J, φ). The classical KAM theorem applies to non-degenerate n-degree-of-freedom systems: If the Hamiltonian has the form H[sub]0[/sub]+εH[sub]1[/sub]+O(ε[sup]2[/sup]) then the determinant of the Hessian of the unperturbed part H[sub]0[/sub] with respect to the action variables must not vanish. This yields that the so called frequency map ist a local diffeomorphism and all those n-tori whose frequencies satisfy a certain Diophantine condition survive small perturbations. The persisting n-tori form a set whose complement has measure O(α).The Hamiltonian H= K+ε H[sub]1[/sub]+ε[sup]2[/sup] H[sub]2[/sub]+hot of the spatial lunar problem is (maximally) superintegrable or properly degenerate. Its unperturbed part, the Kepler Hamiltonian K, depends only on one of three action variables. The classical KAM theorem does not apply. A properly degenerate Hamiltonian H= H[sub]0[/sub]+ε H[sub]1[/sub]+O(&epsilon,[sup]2[/sup]) has a perturbation that removes the degeneracy, if the unperturbed part H[sub]0[/sub] depends on r [ n action variables, is non-degenerate with respect to those and the first order term of the perturbation H[sub]1[/sub] depends non-degenerately on the remaining n-r action variables. For such systems the frequency map of the intermediate system with Hamiltonian H[sub]0[/sub]+ε H[sub]1[/sub] is a diffeomorphism, while the frequencies that occur have two orders of magnitude. For such systems a KAM theorem similar to the classical one exists. The Diophantine condition has to be changed in such a way that it depends linearly on ε. In the case of the lunar problem, the perturbation of the Kepler Hamiltonian does not remove the degeneracy. We have to add the second order term of the perturbation H[sub]2[/sub] to the intermediate Hamiltonian to obtain a dependence on all three action variables. No known KAM theorem applies. But the Kepler Hamiltonian depends only on one action variable and it does so linearly. Using this fact, we derive and prove a new KAM theorem for our class of Hamiltonian systems by adapting the way this has been done for proper degenerate systems whose perturbations remove the degeneracy. Applying this new theorem to the lunar problem yields the persistence of most of the invariant 3-tori of the twice normalized, truncated system in the original one."],"dc:identifier":["https://publications.rwth-aachen.de/record/59205","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-121012%22"],"dc:language":["eng"],"dc:publisher":["Publikationsserver der RWTH Aachen University"],"dc:relation":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-7357","info:eu-repo/semantics/altIdentifier/doi/10.18154/RWTH-CONV-121012"],"dc:rights":["info:eu-repo/semantics/openAccess"],"dc:source":["Aachen : Publikationsserver der RWTH Aachen University 147 S. (2003). doi:10.18154/RWTH-CONV-121012 = Aachen, Techn. Hochsch., Diss., 2003"],"dc:subject":["info:eu-repo/classification/ddc/530","Dreikörperproblem","KAM-Theorie","Physik","Dynamische Systeme","Hamiltonsche Systeme","KAM Theorie","Störungstheorie"],"dc:title":["A KAM theorem for the spatial lunar problem"],"dc:type":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]},"updated_at":"2026-07-30T19:42:39Z"}