# American Institute of Mathematical Sciences

December  2017, 9(4): 487-574. doi: 10.3934/jgm.2017019

## The physical foundations of geometric mechanics

 Department of Mathematics and Statistics, Queen's University, Kingston, ON K7L 3N6, Canada

Research supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.

Received  November 2015 Revised  May 2017 Published  October 2017

The principles of geometric mechanics are extended to the physical elements of mechanics, including space and time, rigid bodies, constraints, forces, and dynamics. What is arrived at is a comprehensive and rigorous presentation of basic mechanics, starting with precise formulations of the physical axioms. A few components of the presentation are novel. One is a mathematical presentation of force and torque, providing certain well-known, but seldom clearly exposited, fundamental theorems about force and torque. The classical principles of Virtual Work and Lagrange-d'Alembert are also given clear mathematical statements in various guises and contexts. Another novel facet of the presentation is its derivation of the Euler-Lagrange equations. Standard derivations of the Euler-Lagrange equations from the equations of motion for Newtonian mechanics are typically done for interconnections of particles. Here this is carried out in a coordinate-free rmner for rigid bodies, giving for the first time a direct geometric path from the Newton-Euler equations to the Euler-Lagrange equations in the rigid body setting.

Citation: Andrew D. Lewis. The physical foundations of geometric mechanics. Journal of Geometric Mechanics, 2017, 9 (4) : 487-574. doi: 10.3934/jgm.2017019
##### References:

show all references

##### References:
A rigid transformation with spatial and body frames
Rod with tip constrained to move in a plane
Central torque-force on a rigid body in a configuration
 [1] Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577 [2] Menita Carozza, Jan Kristensen, Antonia Passarelli di Napoli. On the validity of the Euler-Lagrange system. Communications on Pure & Applied Analysis, 2015, 14 (1) : 51-62. doi: 10.3934/cpaa.2015.14.51 [3] Giovanni Bonfanti, Arrigo Cellina. The validity of the Euler-Lagrange equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 511-517. doi: 10.3934/dcds.2010.28.511 [4] Stefano Bianchini. On the Euler-Lagrange equation for a variational problem. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 449-480. doi: 10.3934/dcds.2007.17.449 [5] Yongcai Geng. Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force. Communications on Pure & Applied Analysis, 2015, 14 (2) : 549-564. doi: 10.3934/cpaa.2015.14.549 [6] Dmitry V. Zenkov, Anthony M. Bloch. Dynamics of generalized Euler tops with constraints. Conference Publications, 2001, 2001 (Special) : 398-405. doi: 10.3934/proc.2001.2001.398 [7] Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987 [8] Renjun Duan, Shuangqian Liu. Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinetic & Related Models, 2013, 6 (4) : 687-700. doi: 10.3934/krm.2013.6.687 [9] Bernard Bonnard, Olivier Cots, Nataliya Shcherbakova. The Serret-Andoyer Riemannian metric and Euler-Poinsot rigid body motion. Mathematical Control & Related Fields, 2013, 3 (3) : 287-302. doi: 10.3934/mcrf.2013.3.287 [10] Paola Goatin, Philippe G. LeFloch. $L^1$ continuous dependence for the Euler equations of compressible fluids dynamics. Communications on Pure & Applied Analysis, 2003, 2 (1) : 107-137. doi: 10.3934/cpaa.2003.2.107 [11] Okihiro Sawada. Analytic rates of solutions to the Euler equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1409-1415. doi: 10.3934/dcdss.2013.6.1409 [12] Luigi Ambrosio. Variational models for incompressible Euler equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 1-10. doi: 10.3934/dcdsb.2009.11.1 [13] Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible Euler-Maxwell equations. Communications on Pure & Applied Analysis, 2013, 12 (1) : 503-518. doi: 10.3934/cpaa.2013.12.503 [14] Franco Flandoli, Dejun Luo. Euler-Lagrangian approach to 3D stochastic Euler equations. Journal of Geometric Mechanics, 2019, 11 (2) : 153-165. doi: 10.3934/jgm.2019008 [15] Andrey Tsiganov. Integrable Euler top and nonholonomic Chaplygin ball. Journal of Geometric Mechanics, 2011, 3 (3) : 337-362. doi: 10.3934/jgm.2011.3.337 [16] Young-Pil Choi. Compressible Euler equations interacting with incompressible flow. Kinetic & Related Models, 2015, 8 (2) : 335-358. doi: 10.3934/krm.2015.8.335 [17] Philippe G. LeFloch, Seiji Ukai. A symmetrization of the relativistic Euler equations with several spatial variables. Kinetic & Related Models, 2009, 2 (2) : 275-292. doi: 10.3934/krm.2009.2.275 [18] Gianluca Crippa, Elizaveta Semenova, Stefano Spirito. Strong continuity for the 2D Euler equations. Kinetic & Related Models, 2015, 8 (4) : 685-689. doi: 10.3934/krm.2015.8.685 [19] Meixiang Huang, Zhi-Qiang Shao. Riemann problem for the relativistic generalized Chaplygin Euler equations. Communications on Pure & Applied Analysis, 2016, 15 (1) : 127-138. doi: 10.3934/cpaa.2016.15.127 [20] A. Alexandrou Himonas, Gerard Misiołek, Feride Tiǧlay. On unique continuation for the modified Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 515-529. doi: 10.3934/dcds.2007.19.515

2018 Impact Factor: 0.525