Deriving Euler’s Equation for Rigid-Body Rotation via Lagrangian Dynamics with Generalized Coordinates
Euler’s equation relates the change in angular momentum of a rigid body to the applied torque. This paper uses Lagrangian dynamics to derive Euler’s equation in terms of generalized coordinates. This is done by parameterizing the angular velocity vector in terms of 3-2-1 and 3-1-3 Euler angles as we...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/12/2727 |
| _version_ | 1797593661910810624 |
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| author | Dennis S. Bernstein Ankit Goel Omran Kouba |
| author_facet | Dennis S. Bernstein Ankit Goel Omran Kouba |
| author_sort | Dennis S. Bernstein |
| collection | DOAJ |
| description | Euler’s equation relates the change in angular momentum of a rigid body to the applied torque. This paper uses Lagrangian dynamics to derive Euler’s equation in terms of generalized coordinates. This is done by parameterizing the angular velocity vector in terms of 3-2-1 and 3-1-3 Euler angles as well as Euler parameters, that is, quaternions. This paper fills a gap in the literature by using generalized coordinates to parameterize the angular velocity vector and thereby transform the dynamics obtained from Lagrangian dynamics into Euler’s equation for rigid-body rotation. |
| first_indexed | 2024-03-11T02:11:30Z |
| format | Article |
| id | doaj.art-97707b73b9ab4e5283b7842fd3943581 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T02:11:30Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-97707b73b9ab4e5283b7842fd39435812023-11-18T11:28:56ZengMDPI AGMathematics2227-73902023-06-011112272710.3390/math11122727Deriving Euler’s Equation for Rigid-Body Rotation via Lagrangian Dynamics with Generalized CoordinatesDennis S. Bernstein0Ankit Goel1Omran Kouba2Aerospace Engineering Department, University of Michigan, Ann Arbor, MI 48109, USAMechanical Engineering Department, University of Maryland, Baltimore County, MD 20742, USADepartment of Mathematics, Higher Institute for Applied Sciences and Technology, Damascus 31983, SyriaEuler’s equation relates the change in angular momentum of a rigid body to the applied torque. This paper uses Lagrangian dynamics to derive Euler’s equation in terms of generalized coordinates. This is done by parameterizing the angular velocity vector in terms of 3-2-1 and 3-1-3 Euler angles as well as Euler parameters, that is, quaternions. This paper fills a gap in the literature by using generalized coordinates to parameterize the angular velocity vector and thereby transform the dynamics obtained from Lagrangian dynamics into Euler’s equation for rigid-body rotation.https://www.mdpi.com/2227-7390/11/12/2727angular velocityrotationquaternions |
| spellingShingle | Dennis S. Bernstein Ankit Goel Omran Kouba Deriving Euler’s Equation for Rigid-Body Rotation via Lagrangian Dynamics with Generalized Coordinates Mathematics angular velocity rotation quaternions |
| title | Deriving Euler’s Equation for Rigid-Body Rotation via Lagrangian Dynamics with Generalized Coordinates |
| title_full | Deriving Euler’s Equation for Rigid-Body Rotation via Lagrangian Dynamics with Generalized Coordinates |
| title_fullStr | Deriving Euler’s Equation for Rigid-Body Rotation via Lagrangian Dynamics with Generalized Coordinates |
| title_full_unstemmed | Deriving Euler’s Equation for Rigid-Body Rotation via Lagrangian Dynamics with Generalized Coordinates |
| title_short | Deriving Euler’s Equation for Rigid-Body Rotation via Lagrangian Dynamics with Generalized Coordinates |
| title_sort | deriving euler s equation for rigid body rotation via lagrangian dynamics with generalized coordinates |
| topic | angular velocity rotation quaternions |
| url | https://www.mdpi.com/2227-7390/11/12/2727 |
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