[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82689-en":3,"doc-seo-82689-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82689,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Variational Integrators Using Forced Discrete Hamiltonian Systems","This paper studies discrete Hamiltonian systems on cotangent bundles subject to external forces, with trajectories defined by a discrete variational principle. The work analyzes how the canonical symplectic structure evolves and how momenta evolve when a Lie symmetry group is present. For continuous forced Hamiltonian dynamics, it builds an exact discrete analogue whose order-r approximations generate trajectories matching the continuous ones with at least order-r accuracy. Two construction methods for approximate discrete systems are provided, leading to a variational integrator via algebraic motion equations.","arXiv :2607 .02694v1 [math .DS] 2 Jul 2026  \nVARIATIONAL INTEGRATORS USING FORCED DISCRETE HAMILTONIAN SYSTEMS  \nMAT´IAS I. CARUSO, JAVIER FERN´ANDEZ, CORA TORI, AND MARCELA ZUCCALLI  \nAbstract. We study discrete Hamiltonian systems defined on cotangent bundles that are subjected to external forces, whose trajectories are determined by a discrete variational principle. We analyze the evolution of the canonical symplectic structure and, when a Lie group of symmetries is present, the corresponding evolution of the associated momenta. Given a continuous forced Hamiltonian system, we construct an exact discrete analogue whose order-r approximations yield trajectories that approximate the continuous ones with accuracy of at least order r. We also give two methods to build approximate discrete systems. Combining these, we obtain a variational integrator: first approximate the exact discrete system and then solve the resulting algebraic  \nequations of motion.  \nContents  \n1. Introduction 1  \n2. Forced Hamiltonian systems: variational approach 4  \n3. Forced discrete Hamiltonian systems 6  \n4. Structural properties 10  \n4.1. Relation with Lagrangian systems 10  \n4.2. The canonical symplectic structure 13  \n4.3. The canonical momentum map 15  \n5. Error analysis: discrete exact systems 16  \n6. Error analysis: approximations 19  \n6.1. Contact order of maps 19  \n6.2. Contact order of systems 21  \n6.3. Discretizations of FDHSs 21  \n7. Construction of forced discrete Hamiltonian systems 23  \nAcknowledgments 27  \nReferences 27  \n1. Introduction  \nIn Numerical Analysis, Geometric Numerical Integration refers to the construction of algorithms that approximate the solution of ordinary differential equations while preserving the geometric characteristics of the given problem [HLW06] . When the differential equations arise as equations of motion of mechanical systems there  \n2020 Mathematics Subject Classification. Primary: 37J06, 65P10; Secondary: 70G75 .  \nKey words and phrases. Geometric mechanics, Forced discrete mechanical systems, Hamiltonian systems, Geometric numerical integrator.  \n2 MAT´IAS I. CARUSO, JAVIER FERN´ANDEZ, CORA TORI, AND MARCELA ZUCCALLI is a well known way of constructing geometric integrators using what are known as Discrete Mechanical Systems. The solution of the equation of motion of a (continuous) mechanical system, a trajectory, can be seen as a critical point of a variational problem in a space of paths. Similarly, trajectories of a discrete mechanical system are critical points of a certain functional—defined in a space of discrete paths that, crucially, is finite dimensional—and are characterized by equations of motion that are algebraic. Solving these equations gives rise to a numerical integrator of the original differential problem (see [MW01]), known as a variational integrator, provided that  \n(1) the discrete mechanical system is “close enough” to the continuous one, and  \n(2) that (1) suffices to conclude that the trajectories of the systems are “close enough”.  \nAlso of importance,  \n(3) how well do variational integrators preserve the geometric characteristics of the system?  \nThe description of trajectories of mechanical systems defined on a configuration space Q as critical points of a functional is characteristic of the Lagrangian formulation of Mechanics—variational formulation would be a better name—, where the functional, called the action, is computed using a Lagrangian function L over the tangent bundle TQ. Alternatively, it is possible to give a characterization of trajectories as critical points of a functional on curves in the cotangent bundle T∗ Q, computed using a Hamiltonian function H on T∗ Q (see [AM78] or [Gol80]) . In most cases, the two descriptions are equivalent. Both approaches have discrete versions: by far, the most common is the Lagrangian approach, where the discrete action is defined using a discrete Lagrangian function Ld : Q×Q → R (see [WM97],[MW01] and [MMM06]) . The discrete ","cbCaiiaIEoEyv2Mz","https://ap.wps.com/l/cbCaiiaIEoEyv2Mz","pdf",3862188,1,28,"English","en",105,"# Introduction\n# Forced Hamiltonian systems: variational approach\n# Forced discrete Hamiltonian systems\n# Structural properties\n## Relation with Lagrangian systems\n## The canonical symplectic structure\n## The canonical momentum map\n# Error analysis: discrete exact systems\n# Error analysis: approximations\n## Contact order of maps\n## Contact order of systems\n## Discretizations of FDHSs\n# Construction of forced discrete Hamiltonian systems","[{\"question\":\"What defines the trajectories of the forced discrete Hamiltonian systems in this work?\",\"answer\":\"Trajectories are defined as solutions to a discrete variational problem, yielding equations of motion that are algebraic.\"},{\"question\":\"How does the paper treat symmetries in forced discrete Hamiltonian systems?\",\"answer\":\"When a Lie group of symmetries is present, it studies the evolution of the associated canonical momenta alongside the symplectic structure evolution.\"},{\"question\":\"How are continuous forced Hamiltonian systems related to the discrete models constructed here?\",\"answer\":\"An exact discrete analogue is constructed from a continuous forced Hamiltonian system, and order-r approximations produce discrete trajectories that approximate the continuous ones with at least order-r accuracy.\"}]",1784182304,71,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"variational-integrators-using-forced-discrete-hamiltonian-systems","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/variational-integrators-using-forced-discrete-hamiltonian-systems/82689/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What defines the trajectories of the forced discrete Hamiltonian systems in this work?","Question",{"text":75,"@type":76},"Trajectories are defined as solutions to a discrete variational problem, yielding equations of motion that are algebraic.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper treat symmetries in forced discrete Hamiltonian systems?",{"text":80,"@type":76},"When a Lie group of symmetries is present, it studies the evolution of the associated canonical momenta alongside the symplectic structure evolution.",{"name":82,"@type":73,"acceptedAnswer":83},"How are continuous forced Hamiltonian systems related to the discrete models constructed here?",{"text":84,"@type":76},"An exact discrete analogue is constructed from a continuous forced Hamiltonian system, and order-r approximations produce discrete trajectories that approximate the continuous ones with at least order-r accuracy.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]