Canonical transformations of local functionals and sh-Lie structures |
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| Affiliation: | 1. Department of Polymer Engineering & Color Technology, Amirkabir University of Technology, Tehran, Iran;2. Catalysis and Nanomaterials Research Laboratory, School of Chemical, Petroleum, and Gas Engineering, Iran University of Science and Technology, Tehran, Iran;3. School of Chemical Engineering, College of Engineering, University of Tehran;4. Department of Chemical Technologies, Iranian Research Organization for Science and Technology (IROST), P.O. Box 33535111, Tehran, Iran;5. Mineral Recovery Research Center (MRRC), School of Engineering, Edith Cowan University, Joondalup, Perth, WA 6027, Australia;6. Khalifa University, Department of Chemical Engineering, Abu Dhabi, United Arab Emirates;7. Department of Chemical and Materials Engineering, Faculty of Engineering, University of Auckland, Auckland, New Zealand;8. Circular Innovations (CIRCUIT) Research Centre, The University of Auckland, Auckland 1010, New Zealand;9. NGĀ ARA WHETŪ Centre for Climate, Biodiversity and Society, The University of Auckland, Auckland 1010, New Zealand;10. UNESCO Centre for Membrane Science and Technology, School of Chemical Engineering, University of New South Wales, Sydney, NSW 2052 Australia;1. School of Civil Engineering, Guangzhou University, PR China;2. The National Centre for Building and Construction Technology, King Abdulaziz City for Science and Technology, Kingdom of Saudi Arabia;3. School of Civil Engineering, University of Queensland, 4072, Australia;4. Slovenian National Building and Civil Engineering Institute (ZAG), Dimičeva ulica 12, 1000 Ljubljana, Slovenia;5. School of Architecture and Civil Engineering, Xiamen University, Xiamen, Fujian, PR China;1. Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran;2. Computer Science Department, Seattle University, Seattle 98122 USA;3. Department of Electrical and Computer Engineering, McMaster University, L8S 4L8, Canada |
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| Abstract: | In many Lagrangian field theories, there is a Poisson bracket on the space of local functionals. One may identify the fields of such theories as sections of a vector bundle. It is known that the Poisson bracket induces an sh-Lie structure on the graded space of horizontal forms on the jet bundle of the relevant vector bundle. We consider those automorphisms of the vector bundle which induce mappings on the space of functionals preserving the Poisson bracket and refer to such automorphisms as canonical automorphisms.We determine how such automorphisms relate to the corresponding sh-Lie structure. If a Lie group acts on the bundle via canonical automorphisms, there are induced actions on the space of local functionals and consequently on the corresponding sh-Lie algebra. We determine conditions under which the sh-Lie structure induces an sh-Lie structure on a corresponding reduced space where the reduction is determined by the action of the group. These results are not directly a consequence of the corresponding theorems on Poisson manifolds as none of the algebraic structures are Poisson algebras. |
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