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

The physical foundations of geometric mechanics

Andrew D. Lewis

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.

Keywords: Rigid body dynamics, force and torque, Newton-Euler equations, Euler-Lagrange equations, nonholonomic constraints.
Mathematics Subject Classification: Primary: 70E55; Secondary: 70G45, 70G65, 70G75, 70H03.
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.  Google Scholar

[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.  Google Scholar

[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].  Google Scholar

[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]. Google Scholar

[5]

R. E. Artz, Classical mechanics in Galilean space-time, Foundations of Physics, 11 (1981), 679-697.  doi: 10.1007/BF00726944.  Google Scholar

[6]

M. Berger, Geometry I, Universitext, Springer-Verlag, New York/Heidelberg/Berlin, 1987. doi: 10.1007/978-3-540-93815-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

B. Brogliato, Nonsmooth Mechanics, Third edition. Communications and Control Engineering Series. Springer, [Cham], 2016. doi: 10.1007/978-3-319-28664-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

D. L. Cohn, Measure Theory, 2nd edition, Birkhäuser Advanced Texts, Birkhäuser, Boston/Basel/Stuttgart, 2013. doi: 10.1007/978-1-4614-6956-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

M. R. Flannery, The enigma of nonholonomic constraints, American Journal of Physics, 73 (2005), 265-272.  doi: 10.1119/1.1830501.  Google Scholar

[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.  Google Scholar

[17]

H. Goldstein, Classical Mechanics, Addison Wesley, Reading, MA, 1951, New edition: [18].  Google Scholar

[18]

H. Goldstein, C. P. Poole, Jr and J. L. Safko, Classical Mechanics, 3rd edition, Addison Wesley, Reading, MA, 2001, Original edition: [17]. Google Scholar

[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.  Google Scholar

[20]

A. Hatcher, Algebraic Topology, Cambridge University Press, New York/Port Chester/Melbourne/Sydney, 2002. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

Y. Kanno, Nonsmooth Mechanics and Convex Optimization, CRC Press, Boca Raton, FL, 2011. doi: 10.1201/b10839.  Google Scholar

[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.  Google Scholar

[25]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969.  Google Scholar

[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.  Google Scholar

[27]

J. L. Lagrange, Méchanique Analitique, Chez la Veuve Desaint, Paris, 1788, Translation: [28]. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

A. D. Lewis, Is it worth learning differential geometric methods for modelling and control of mechanical systems?, Robotica, 26 (2007), 765-777.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

J. E. Marsden and T. R. Hughes, Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, NJ, 1983, Reprint: [35]. Google Scholar

[35]

J. E. Marsden and T. R. Hughes, Mathematical Foundations of Elasticity, Dover Publications, Inc., New York, 1994, Original edition: [34].  Google Scholar

[36]

J. E. Marsden and J. Scheurle, The reduced Euler-Lagrange equations, Fields Institute Communications, Fields Institute, 1 (1993), 139-164.   Google Scholar

[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.  Google Scholar

[38]

R. M. Murray, Z. X. Li and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, 1994.  Google Scholar

[39]

O. M. O'Reilly, Intermediate Dynamics for Engineers, Cambridge University Press, New York/Port Chester/Melbourne/Sydney, 2008. doi: 10.1017/CBO9780511791352.  Google Scholar

[40]

J. G. Papastavridis, Tensor Calculus and Analytical Dynamics, Library of Engineering Mathematics, CRC Press, Boca Raton, FL, 1999.  Google Scholar

[41]

L. A. Pars, A Treatise on Analytical Dynamics, John Wiley and Sons, New York, 1965.  Google Scholar

[42]

M. Spivak, Physics for Mathematicians. Mechanics I, Publish or Perish, Inc., Houston, 2010.  Google Scholar

[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].  Google Scholar

[44]

C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, Springer-Verlag, New York/Heidelberg/Berlin, 2004, Original edition: [43]. Google Scholar

[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].  Google Scholar

[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.  Google Scholar

show all references

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

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd edition, Addison Wesley, Reading, MA, 1978.  Google Scholar

[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.  Google Scholar

[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].  Google Scholar

[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]. Google Scholar

[5]

R. E. Artz, Classical mechanics in Galilean space-time, Foundations of Physics, 11 (1981), 679-697.  doi: 10.1007/BF00726944.  Google Scholar

[6]

M. Berger, Geometry I, Universitext, Springer-Verlag, New York/Heidelberg/Berlin, 1987. doi: 10.1007/978-3-540-93815-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

B. Brogliato, Nonsmooth Mechanics, Third edition. Communications and Control Engineering Series. Springer, [Cham], 2016. doi: 10.1007/978-3-319-28664-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

D. L. Cohn, Measure Theory, 2nd edition, Birkhäuser Advanced Texts, Birkhäuser, Boston/Basel/Stuttgart, 2013. doi: 10.1007/978-1-4614-6956-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

M. R. Flannery, The enigma of nonholonomic constraints, American Journal of Physics, 73 (2005), 265-272.  doi: 10.1119/1.1830501.  Google Scholar

[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.  Google Scholar

[17]

H. Goldstein, Classical Mechanics, Addison Wesley, Reading, MA, 1951, New edition: [18].  Google Scholar

[18]

H. Goldstein, C. P. Poole, Jr and J. L. Safko, Classical Mechanics, 3rd edition, Addison Wesley, Reading, MA, 2001, Original edition: [17]. Google Scholar

[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.  Google Scholar

[20]

A. Hatcher, Algebraic Topology, Cambridge University Press, New York/Port Chester/Melbourne/Sydney, 2002. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

Y. Kanno, Nonsmooth Mechanics and Convex Optimization, CRC Press, Boca Raton, FL, 2011. doi: 10.1201/b10839.  Google Scholar

[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.  Google Scholar

[25]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969.  Google Scholar

[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.  Google Scholar

[27]

J. L. Lagrange, Méchanique Analitique, Chez la Veuve Desaint, Paris, 1788, Translation: [28]. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

A. D. Lewis, Is it worth learning differential geometric methods for modelling and control of mechanical systems?, Robotica, 26 (2007), 765-777.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

J. E. Marsden and T. R. Hughes, Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, NJ, 1983, Reprint: [35]. Google Scholar

[35]

J. E. Marsden and T. R. Hughes, Mathematical Foundations of Elasticity, Dover Publications, Inc., New York, 1994, Original edition: [34].  Google Scholar

[36]

J. E. Marsden and J. Scheurle, The reduced Euler-Lagrange equations, Fields Institute Communications, Fields Institute, 1 (1993), 139-164.   Google Scholar

[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.  Google Scholar

[38]

R. M. Murray, Z. X. Li and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, 1994.  Google Scholar

[39]

O. M. O'Reilly, Intermediate Dynamics for Engineers, Cambridge University Press, New York/Port Chester/Melbourne/Sydney, 2008. doi: 10.1017/CBO9780511791352.  Google Scholar

[40]

J. G. Papastavridis, Tensor Calculus and Analytical Dynamics, Library of Engineering Mathematics, CRC Press, Boca Raton, FL, 1999.  Google Scholar

[41]

L. A. Pars, A Treatise on Analytical Dynamics, John Wiley and Sons, New York, 1965.  Google Scholar

[42]

M. Spivak, Physics for Mathematicians. Mechanics I, Publish or Perish, Inc., Houston, 2010.  Google Scholar

[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].  Google Scholar

[44]

C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, Springer-Verlag, New York/Heidelberg/Berlin, 2004, Original edition: [43]. Google Scholar

[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].  Google Scholar

[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.  Google Scholar

Figure 1.  A rigid transformation with spatial and body frames
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Figure 2.  Rod with tip constrained to move in a plane
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Figure 3.  Central torque-force on a rigid body in a configuration
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