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We consider three cases, when both Hamiltonians systems in each side of the discontinuity line have simultaneously degree one, two or three. We obtain that in these three cases, this maximum number is zero, one and three, respectively. Moreover, we prove that there are discontinuous piecewise differential systems realizing these maximum number of limit cycles. Note that we have solved the extension of the 16th Hilbert problem about the maximum number of limit cycles that these three classes of discontinuous piecewise differential systems separated by one straight line and formed by two Hamiltonian systems with a degree either one, two, or three, which such systems can exhibit.<\/jats:p>","DOI":"10.3390\/axioms13030161","type":"journal-article","created":{"date-parts":[[2024,2,29]],"date-time":"2024-02-29T02:59:39Z","timestamp":1709175579000},"page":"161","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Limit Cycles of Discontinuous Piecewise Differential Hamiltonian Systems Separated by a Straight Line"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3870-978X","authenticated-orcid":false,"given":"Joyce A.","family":"Casimiro","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica, Universidade Estadual de Campinas, Cidade Universit\u00e1ria Zeferino Vaz, Rua S\u00e9rgio Baruque de Holanda, 651, Campinas 13083-859, SP, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9511-5999","authenticated-orcid":false,"given":"Jaume","family":"Llibre","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat Aut\u00f2noma de Barcelona, 08193 Bellaterra, Catalonia, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Appell, J., Hien, N.T., Petrova, L., and Pryadko, I. 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