# 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:
 [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd edition, Addison Wesley, Reading, MA, 1978. [2] R. Abraham, J. E. Marsden and T. S. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd edition, no. 75 in Applied Mathematical Sciences, Springer-Verlag, New York/Heidelberg/Berlin, 1988. doi: 10.1007/978-1-4612-1029-0. [3] V. I. Arnol'ed, Mathematical Methods of Classical Mechanics, 2nd edition, no. 60 in Graduate Texts in Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 1978, New edition: [4]. [4] V. I. Arnol'd, Mathematical Methods of Classical Mechanics, 2nd edition, no. 60 in Graduate Texts in Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 1989, First edition: [3]. [5] R. E. Artz, Classical mechanics in Galilean space-time, Foundations of Physics, 11 (1981), 679-697. doi: 10.1007/BF00726944. [6] M. Berger, Geometry I, Universitext, Springer-Verlag, New York/Heidelberg/Berlin, 1987. doi: 10.1007/978-3-540-93815-6. [7] A. Bhand and A. D. Lewis, Rigid body mechanics in Galilean spacetimes Journal of Mathematical Physics, 46 (2005), 102902, 29 pp. doi: 10.1063/1.2060547. [8] A. M. Bloch, Nonholonomic Mechanics and Control, no. 24 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 2003. doi: 10.1007/b97376. [9] B. Brogliato, Nonsmooth Mechanics, Third edition. Communications and Control Engineering Series. Springer, [Cham], 2016. doi: 10.1007/978-3-319-28664-8. [10] F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Systems, no. 49 in Texts in Applied Mathematics. Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7. [11] H. Cendra, D. D. Holm, J. E. Marsden and T. S. Ratiu, Lagrangian reduction, the Euler-Poincaré equations, and semidirect products, American Mathematical Society Translations, Series 2, 186 (1998), 1-25. doi: 10.1090/trans2/186/01. [12] D. L. Cohn, Measure Theory, 2nd edition, Birkhäuser Advanced Texts, Birkhäuser, Boston/Basel/Stuttgart, 2013. doi: 10.1007/978-1-4614-6956-8. [13] J. Cortés, M. de León, D. M. de Diego and S. Martínez, Mechanical systems subjected to generalized non-holonomic constraints, Proceedings of the Royal Society. London. Series A. Mathematical and Physical Sciences, 457 (2001), 651-670. doi: 10.1098/rspa.2000.0686. [14] M. Crampin, On the concept of angular velocity, European Journal of Physics, 7 (1986), 287-293. doi: 10.1088/0143-0807/7/4/014. [15] M. R. Flannery, The enigma of nonholonomic constraints, American Journal of Physics, 73 (2005), 265-272. doi: 10.1119/1.1830501. [16] C. Glocker, Set Valued Force Laws, no. 1 in Lecture Notes in Applied Mechanics, Springer-Verlag, New York/Heidelberg/Berlin, 2001. doi: 10.1007/978-3-540-44479-4. [17] H. Goldstein, Classical Mechanics, Addison Wesley, Reading, MA, 1951, New edition: [18]. [18] H. Goldstein, C. P. Poole, Jr and J. L. Safko, Classical Mechanics, 3rd edition, Addison Wesley, Reading, MA, 2001, Original edition: [17]. [19] X. Grácia, J. Marin-Solano and M.-C. Muñoz-Lecanda, Some geometric aspects of variational calculus in constrained systems, Reports on Mathematical Physics, 51 (2003), 127-148. doi: 10.1016/S0034-4877(03)80006-X. [20] A. Hatcher, Algebraic Topology, Cambridge University Press, New York/Port Chester/Melbourne/Sydney, 2002. [21] D. Husemoller, Fibre Bundles, 3rd edition, no. 20 in Graduate Texts in Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 1994. doi: 10.1007/978-1-4757-2261-1. [22] S. Jafarpour and A. D. Lewis, Time-Varying Vector Fields and Their Flows, Springer Briefs in Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 2014. doi: 10.1007/978-3-319-10139-2. [23] Y. Kanno, Nonsmooth Mechanics and Convex Optimization, CRC Press, Boca Raton, FL, 2011. doi: 10.1201/b10839. [24] P. V. Kharlomov, A critique of some mathematical models of mechanical systems with differential constraints, Journal of Applied Mathematics and Mechanics. Translation of the Soviet journal Prikladnaya Matematika i Mekhanika, 56 (1992), 584-594. doi: 10.1016/0021-8928(92)90016-2. [25] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. [26] V. V. Kozlov, The problem of realizing constraints in dynamics, Journal of Applied Mathematics and Mechanics. Translation of the Soviet journal Prikladnaya Matematika i Mekhanika, 56 (1992), 594-600. doi: 10.1016/0021-8928(92)90017-3. [27] J. L. Lagrange, Méchanique Analitique, Chez la Veuve Desaint, Paris, 1788, Translation: [28]. [28] J. L. Lagrange, Analytical Mechanics, No. 191 in Boston Studies in the Philosophy of Science, Kluwer Academic Publishers, Dordrecht, 1997, Original edition: [27]. doi: 10.1007/978-94-015-8903-1. [29] A. D. Lewis, The geometry of the Gibbs-Appell equations and Gauss's Principle of Least Constraint, Reports on Mathematical Physics, 38 (1996), 11-28. doi: 10.1016/0034-4877(96)87675-0. [30] A. D. Lewis, Affine connections and distributions with applications to nonholonomic mechanics, Reports on Mathematical Physics, 42 (1998), 135-164. doi: 10.1016/S0034-4877(98)80008-6. [31] A. D. Lewis, Is it worth learning differential geometric methods for modelling and control of mechanical systems?, Robotica, 26 (2007), 765-777. [32] A. D. Lewis and R. M. Murray, Variational principles for constrained systems: Theory and experiment, International Journal of Non-Linear Mechanics, 30 (1995), 793-815. doi: 10.1016/0020-7462(95)00024-0. [33] P. Liberrmn and C. -M. Marle, Symplectic Geometry and Analytical Mechanics, no. 35 in Mathematics and its Applications, D. Reidel Publishing Company, Dordrecht/Boston/Lancaster/Tokyo, 1987. doi: 10.1007/978-94-009-3807-6. [34] J. E. Marsden and T. R. Hughes, Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, NJ, 1983, Reprint: [35]. [35] J. E. Marsden and T. R. Hughes, Mathematical Foundations of Elasticity, Dover Publications, Inc., New York, 1994, Original edition: [34]. [36] J. E. Marsden and J. Scheurle, The reduced Euler-Lagrange equations, Fields Institute Communications, Fields Institute, 1 (1993), 139-164. [37] M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems, No. 9 in Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston/Basel/Stuttgart, 1993. doi: 10.1007/978-3-0348-7614-8. [38] R. M. Murray, Z. X. Li and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, 1994. [39] O. M. O'Reilly, Intermediate Dynamics for Engineers, Cambridge University Press, New York/Port Chester/Melbourne/Sydney, 2008. doi: 10.1017/CBO9780511791352. [40] J. G. Papastavridis, Tensor Calculus and Analytical Dynamics, Library of Engineering Mathematics, CRC Press, Boca Raton, FL, 1999. [41] L. A. Pars, A Treatise on Analytical Dynamics, John Wiley and Sons, New York, 1965. [42] M. Spivak, Physics for Mathematicians. Mechanics I, Publish or Perish, Inc., Houston, 2010. [43] C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, no. 3 in Handbuch der Physik, Springer-Verlag, New York/Heidelberg/Berlin, 1965, New edition: [44]. [44] C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, Springer-Verlag, New York/Heidelberg/Berlin, 2004, Original edition: [43]. [45] E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed. Cambridge University Press, New York, 1959, New edition: [45]. [46] E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge Mathematical Library, Cambridge University Press, New York/Port Chester/Melbourne/Sydney, 1988, Original edition: [45]. doi: 10.1017/CBO9780511608797.

show all references

##### References:
 [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd edition, Addison Wesley, Reading, MA, 1978. [2] R. Abraham, J. E. Marsden and T. S. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd edition, no. 75 in Applied Mathematical Sciences, Springer-Verlag, New York/Heidelberg/Berlin, 1988. doi: 10.1007/978-1-4612-1029-0. [3] V. I. Arnol'ed, Mathematical Methods of Classical Mechanics, 2nd edition, no. 60 in Graduate Texts in Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 1978, New edition: [4]. [4] V. I. Arnol'd, Mathematical Methods of Classical Mechanics, 2nd edition, no. 60 in Graduate Texts in Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 1989, First edition: [3]. [5] R. E. Artz, Classical mechanics in Galilean space-time, Foundations of Physics, 11 (1981), 679-697. doi: 10.1007/BF00726944. [6] M. Berger, Geometry I, Universitext, Springer-Verlag, New York/Heidelberg/Berlin, 1987. doi: 10.1007/978-3-540-93815-6. [7] A. Bhand and A. D. Lewis, Rigid body mechanics in Galilean spacetimes Journal of Mathematical Physics, 46 (2005), 102902, 29 pp. doi: 10.1063/1.2060547. [8] A. M. Bloch, Nonholonomic Mechanics and Control, no. 24 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 2003. doi: 10.1007/b97376. [9] B. Brogliato, Nonsmooth Mechanics, Third edition. Communications and Control Engineering Series. Springer, [Cham], 2016. doi: 10.1007/978-3-319-28664-8. [10] F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Systems, no. 49 in Texts in Applied Mathematics. Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7. [11] H. Cendra, D. D. Holm, J. E. Marsden and T. S. Ratiu, Lagrangian reduction, the Euler-Poincaré equations, and semidirect products, American Mathematical Society Translations, Series 2, 186 (1998), 1-25. doi: 10.1090/trans2/186/01. [12] D. L. Cohn, Measure Theory, 2nd edition, Birkhäuser Advanced Texts, Birkhäuser, Boston/Basel/Stuttgart, 2013. doi: 10.1007/978-1-4614-6956-8. [13] J. Cortés, M. de León, D. M. de Diego and S. Martínez, Mechanical systems subjected to generalized non-holonomic constraints, Proceedings of the Royal Society. London. Series A. Mathematical and Physical Sciences, 457 (2001), 651-670. doi: 10.1098/rspa.2000.0686. [14] M. Crampin, On the concept of angular velocity, European Journal of Physics, 7 (1986), 287-293. doi: 10.1088/0143-0807/7/4/014. [15] M. R. Flannery, The enigma of nonholonomic constraints, American Journal of Physics, 73 (2005), 265-272. doi: 10.1119/1.1830501. [16] C. Glocker, Set Valued Force Laws, no. 1 in Lecture Notes in Applied Mechanics, Springer-Verlag, New York/Heidelberg/Berlin, 2001. doi: 10.1007/978-3-540-44479-4. [17] H. Goldstein, Classical Mechanics, Addison Wesley, Reading, MA, 1951, New edition: [18]. [18] H. Goldstein, C. P. Poole, Jr and J. L. Safko, Classical Mechanics, 3rd edition, Addison Wesley, Reading, MA, 2001, Original edition: [17]. [19] X. Grácia, J. Marin-Solano and M.-C. Muñoz-Lecanda, Some geometric aspects of variational calculus in constrained systems, Reports on Mathematical Physics, 51 (2003), 127-148. doi: 10.1016/S0034-4877(03)80006-X. [20] A. Hatcher, Algebraic Topology, Cambridge University Press, New York/Port Chester/Melbourne/Sydney, 2002. [21] D. Husemoller, Fibre Bundles, 3rd edition, no. 20 in Graduate Texts in Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 1994. doi: 10.1007/978-1-4757-2261-1. [22] S. Jafarpour and A. D. Lewis, Time-Varying Vector Fields and Their Flows, Springer Briefs in Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 2014. doi: 10.1007/978-3-319-10139-2. [23] Y. Kanno, Nonsmooth Mechanics and Convex Optimization, CRC Press, Boca Raton, FL, 2011. doi: 10.1201/b10839. [24] P. V. Kharlomov, A critique of some mathematical models of mechanical systems with differential constraints, Journal of Applied Mathematics and Mechanics. Translation of the Soviet journal Prikladnaya Matematika i Mekhanika, 56 (1992), 584-594. doi: 10.1016/0021-8928(92)90016-2. [25] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. [26] V. V. Kozlov, The problem of realizing constraints in dynamics, Journal of Applied Mathematics and Mechanics. Translation of the Soviet journal Prikladnaya Matematika i Mekhanika, 56 (1992), 594-600. doi: 10.1016/0021-8928(92)90017-3. [27] J. L. Lagrange, Méchanique Analitique, Chez la Veuve Desaint, Paris, 1788, Translation: [28]. [28] J. L. Lagrange, Analytical Mechanics, No. 191 in Boston Studies in the Philosophy of Science, Kluwer Academic Publishers, Dordrecht, 1997, Original edition: [27]. doi: 10.1007/978-94-015-8903-1. [29] A. D. Lewis, The geometry of the Gibbs-Appell equations and Gauss's Principle of Least Constraint, Reports on Mathematical Physics, 38 (1996), 11-28. doi: 10.1016/0034-4877(96)87675-0. [30] A. D. Lewis, Affine connections and distributions with applications to nonholonomic mechanics, Reports on Mathematical Physics, 42 (1998), 135-164. doi: 10.1016/S0034-4877(98)80008-6. [31] A. D. Lewis, Is it worth learning differential geometric methods for modelling and control of mechanical systems?, Robotica, 26 (2007), 765-777. [32] A. D. Lewis and R. M. Murray, Variational principles for constrained systems: Theory and experiment, International Journal of Non-Linear Mechanics, 30 (1995), 793-815. doi: 10.1016/0020-7462(95)00024-0. [33] P. Liberrmn and C. -M. Marle, Symplectic Geometry and Analytical Mechanics, no. 35 in Mathematics and its Applications, D. Reidel Publishing Company, Dordrecht/Boston/Lancaster/Tokyo, 1987. doi: 10.1007/978-94-009-3807-6. [34] J. E. Marsden and T. R. Hughes, Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, NJ, 1983, Reprint: [35]. [35] J. E. Marsden and T. R. Hughes, Mathematical Foundations of Elasticity, Dover Publications, Inc., New York, 1994, Original edition: [34]. [36] J. E. Marsden and J. Scheurle, The reduced Euler-Lagrange equations, Fields Institute Communications, Fields Institute, 1 (1993), 139-164. [37] M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems, No. 9 in Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston/Basel/Stuttgart, 1993. doi: 10.1007/978-3-0348-7614-8. [38] R. M. Murray, Z. X. Li and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, 1994. [39] O. M. O'Reilly, Intermediate Dynamics for Engineers, Cambridge University Press, New York/Port Chester/Melbourne/Sydney, 2008. doi: 10.1017/CBO9780511791352. [40] J. G. Papastavridis, Tensor Calculus and Analytical Dynamics, Library of Engineering Mathematics, CRC Press, Boca Raton, FL, 1999. [41] L. A. Pars, A Treatise on Analytical Dynamics, John Wiley and Sons, New York, 1965. [42] M. Spivak, Physics for Mathematicians. Mechanics I, Publish or Perish, Inc., Houston, 2010. [43] C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, no. 3 in Handbuch der Physik, Springer-Verlag, New York/Heidelberg/Berlin, 1965, New edition: [44]. [44] C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, Springer-Verlag, New York/Heidelberg/Berlin, 2004, Original edition: [43]. [45] E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed. Cambridge University Press, New York, 1959, New edition: [45]. [46] E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge Mathematical Library, Cambridge University Press, New York/Port Chester/Melbourne/Sydney, 1988, Original edition: [45]. doi: 10.1017/CBO9780511608797.
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
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