Noether’s theorem for nonconservative systems in quasicoordinates
In this paper the generalized Noether’s theorem is given in quasicoordinates for the systems of particles, the motion of which can be presented in quasicoordinats and quasivelocities. After a systematic review of the calculus with quasicoordinates and the corresponding Boltzmann-Hamel’s...
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| Format: | Article |
| Language: | English |
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Serbian Society of Mechanics & Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade
2016-01-01
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| Series: | Theoretical and Applied Mechanics |
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| Online Access: | http://www.doiserbia.nb.rs/img/doi/1450-5584/2016/1450-55841600004M.pdf |
| _version_ | 1818550260787576832 |
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| author | Mušicki Đorđe |
| author_facet | Mušicki Đorđe |
| author_sort | Mušicki Đorđe |
| collection | DOAJ |
| description | In this paper the generalized Noether’s theorem is given in quasicoordinates
for the systems of particles, the motion of which can be presented in
quasicoordinats and quasivelocities. After a systematic review of the
calculus with quasicoordinates and the corresponding Boltzmann-Hamel’s
equations of motion, the total variation of action is given in
quasicoordinates. Then, the corresponding generalized Noether’s theorem is
formulated, valid for nonconservative systems as well, which is obtained
from the total variation of action and corresponding Boltzmann-Hamel’s
equations. So formulated Noether’s theoerm in quasicoordinates is valid for
all conservative and nonconservative systems without any limitation. It is
applied to obtain the corresponding energy integrals in quasicoordinates for
conservative and nonconservative systems, in the latter case these are
energy integrals in broader sense. The obtained results are illustrated by a
characteristic example, where the corresponding energy integral is found.
This generalized Neother’s theorem is equivalent, but not in the form and
with some limitation, to the corresponding Noether’s theorem formulated by
Dj. Djuki.c [13], which is obtained from the invariance of total variation
only of element of action Δ(𝐿𝑑𝑡). However, for nonconservative systems
the Lagrangian 𝐿, appearing in this relations, represents not the usual,
but an equivalent Lagrangian, which completely determines the considered
system, including the influence of nonpotential forces. Therefore, the cited
Noether’s theorem is valid only for these nonconservative systems for which
it is possible to find such equivalent Lagrangian, (what for the natural
systems is mostly possible). |
| first_indexed | 2024-12-12T08:44:05Z |
| format | Article |
| id | doaj.art-55ddac11f32940a7937eff8982c94d19 |
| institution | Directory Open Access Journal |
| issn | 1450-5584 2406-0925 |
| language | English |
| last_indexed | 2024-12-12T08:44:05Z |
| publishDate | 2016-01-01 |
| publisher | Serbian Society of Mechanics & Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade |
| record_format | Article |
| series | Theoretical and Applied Mechanics |
| spelling | doaj.art-55ddac11f32940a7937eff8982c94d192022-12-22T00:30:39ZengSerbian Society of Mechanics & Mathematical Institute of the Serbian Academy of Sciences and Arts, BelgradeTheoretical and Applied Mechanics1450-55842406-09252016-01-0143111710.2298/TAM151020004M1450-55841600004MNoether’s theorem for nonconservative systems in quasicoordinatesMušicki Đorđe0University of Belgrade, Faculty of Physic, Belgrade, Serbia + Mathematical Institute SANU, Belgrade, SerbiaIn this paper the generalized Noether’s theorem is given in quasicoordinates for the systems of particles, the motion of which can be presented in quasicoordinats and quasivelocities. After a systematic review of the calculus with quasicoordinates and the corresponding Boltzmann-Hamel’s equations of motion, the total variation of action is given in quasicoordinates. Then, the corresponding generalized Noether’s theorem is formulated, valid for nonconservative systems as well, which is obtained from the total variation of action and corresponding Boltzmann-Hamel’s equations. So formulated Noether’s theoerm in quasicoordinates is valid for all conservative and nonconservative systems without any limitation. It is applied to obtain the corresponding energy integrals in quasicoordinates for conservative and nonconservative systems, in the latter case these are energy integrals in broader sense. The obtained results are illustrated by a characteristic example, where the corresponding energy integral is found. This generalized Neother’s theorem is equivalent, but not in the form and with some limitation, to the corresponding Noether’s theorem formulated by Dj. Djuki.c [13], which is obtained from the invariance of total variation only of element of action Δ(𝐿𝑑𝑡). However, for nonconservative systems the Lagrangian 𝐿, appearing in this relations, represents not the usual, but an equivalent Lagrangian, which completely determines the considered system, including the influence of nonpotential forces. Therefore, the cited Noether’s theorem is valid only for these nonconservative systems for which it is possible to find such equivalent Lagrangian, (what for the natural systems is mostly possible).http://www.doiserbia.nb.rs/img/doi/1450-5584/2016/1450-55841600004M.pdfNoether’s theoremquasicoordinates and quasivelocitiesenergy integrals (in usual and broader sense) |
| spellingShingle | Mušicki Đorđe Noether’s theorem for nonconservative systems in quasicoordinates Theoretical and Applied Mechanics Noether’s theorem quasicoordinates and quasivelocities energy integrals (in usual and broader sense) |
| title | Noether’s theorem for nonconservative systems in quasicoordinates |
| title_full | Noether’s theorem for nonconservative systems in quasicoordinates |
| title_fullStr | Noether’s theorem for nonconservative systems in quasicoordinates |
| title_full_unstemmed | Noether’s theorem for nonconservative systems in quasicoordinates |
| title_short | Noether’s theorem for nonconservative systems in quasicoordinates |
| title_sort | noether s theorem for nonconservative systems in quasicoordinates |
| topic | Noether’s theorem quasicoordinates and quasivelocities energy integrals (in usual and broader sense) |
| url | http://www.doiserbia.nb.rs/img/doi/1450-5584/2016/1450-55841600004M.pdf |
| work_keys_str_mv | AT musickiđorđe noetherstheoremfornonconservativesystemsinquasicoordinates |