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doi: 10.3934/jgm.2021002

The principle of virtual work and Hamilton's principle on Galilean manifolds

Institute for Nonlinear Mechanics, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany

* Corresponding author: Giuseppe Capobianco

Received  March 2020 Revised  September 2020 Published  January 2021

To describe time-dependent finite-dimensional mechanical systems, their generalized space-time is modeled as a Galilean manifold. On this basis, we present a geometric mechanical theory that unifies Lagrangian and Hamiltonian mechanics. Moreover, a general definition of force is given, such that the theory is capable of treating nonpotential forces acting on a mechanical system. Within this theory, we elaborate the interconnections between classical equations known from analytical mechanics such as the principle of virtual work, Lagrange's equations of the second kind, Hamilton's equations, Lagrange's central equation, Hamel's generalized central equation as well as Hamilton's principle.

Citation: Giuseppe Capobianco, Tom Winandy, Simon R. Eugster. The principle of virtual work and Hamilton's principle on Galilean manifolds. Journal of Geometric Mechanics, doi: 10.3934/jgm.2021002
References:
[1]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[2]

H. Bremer, Dynamik und Regelung mechanischer Systeme, Leitfäden der angewandten Mathematik und Mechanik, 67, Vieweg+Teubner Verlag, Weisbaden, 1988. doi: 10.1007/978-3-663-05674-4.  Google Scholar

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H. Bremer, Elastic Multibody Dynamics. A Direct Ritz Approach, Intelligent Systems, Control and Automation: Science and Engineering, 35, Springer, New York, 2008. doi: 10.1007/978-1-4020-8680-9.  Google Scholar

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É. Cartan, Leçons sur les Invariants Intégraux, Hermann, Paris, 1971.  Google Scholar

[5]

S. R. EugsterG. Capobianco and T. Winandy, Geometric description of time-dependent finite-dimensional mechanical systems, Math. Mech. Solids, 25 (2020), 2050-2075.  doi: 10.1177/1081286520918900.  Google Scholar

[6]

H. Goldstein, Classical Mechanics, Addison-Wesley Series in Physics, Addison-Wesley Publishing Co., Reading, Mass., 1980.  Google Scholar

[7]

G. Hamel, Die Lagrange-Eulerschen Gleichungen der Mechanik, Zeitschrift für Mathematik und Physik, 50 (1904), 1-57.   Google Scholar

[8]

G. Hamel, Theoretische Mechanik, Grundlehren der Mathematischen Wissenschaften, 57, Springer-Verlag, Berlin-New York, 1978.  Google Scholar

[9]

G. Hamel, Über die virtuellen Verschiebungen in der Mechanik, Math. Ann., 59 (1904), 416-434.  doi: 10.1007/BF01445152.  Google Scholar

[10]

W. R. Hamilton, Ⅶ. Second essay on a general method in dynamics, Philos. Transac. Roy. Soc. London, 125 (1835), 95-144.  doi: 10.1098/rstl.1835.0009.  Google Scholar

[11]

R. Hermann, Differential form methods in the theory of variational systems and Lagrangian field theories, Acta Appl. Math., 12 (1988), 35-78.  doi: 10.1007/BF00047568.  Google Scholar

[12]

J. Jost, Riemannian Geometry and Geometric Analysis, Universitext, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-77341-2.  Google Scholar

[13]

J.-L. Lagrange, Théorie de la libration de la lune, Nouv. Mem. Acad. R. Sci. Bruxelles, (1780). Google Scholar

[14]

C. Lánczos, The Variational Principles of Mechanics, Mathematical Expositions, 4, University of Toronto Press, Toronto, Ont., 1949.  Google Scholar

[15]

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 2, The Classical Theory of Fields, Pergamon Press, Oxford-New York-Toronto, Ont., 1975.  Google Scholar

[16]

L. D. Landau and E. M. Lifshitz, Mechanics. Course of Theoretical Physics, Vol. 1, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960.  Google Scholar

[17]

J. M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, 218, Springer, New York, 2013. doi: 10.1007/978-1-4419-9982-5.  Google Scholar

[18]

J. M. Lee, Manifolds and Differential Geometry, Graduate Studies in Mathematics, 107, American Mathematical Society, Providence, RI, 2009. doi: 10.1090/gsm/107.  Google Scholar

[19]

O. Loos, Analytische Mechanik, Seminarausarbeitung, Institut für Mathematik, Universität Innsbruck, 1982. Google Scholar

[20]

O. Loos, Automorphism groups of classical mechanical systems, Monatsh. Math., 100 (1985), 277-292.  doi: 10.1007/BF01339229.  Google Scholar

[21]

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

[22]

J.-M. Souriau, Structure des Systèmes Dynamiques, Dunod, Paris, 1970.  Google Scholar

[23]

J.-M. Souriau, Structure of Dynamical Systems, Progress in Mathematics, 149, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-1-4612-0281-3.  Google Scholar

[24]

M. Spivak, A Comprehensive Introduction to Differential Geometry. Vol. I, Publish or Perish, Inc., Wilmington, Del., 1979.  Google Scholar

[25]

J. L. Synge, Classical Dynamics, Handbuch der Physik, Bd. III/1, Springer, Berlin, 1960, 1–225.  Google Scholar

[26]

T. Winandy, Dynamics of Finite-Dimensional Mechanical Systems, Ph.D thesis, University of Stuttgart, 2019. Google Scholar

[27]

K. Yano and S. Ishihara, Tangent and Cotangent Bundles: Differential Geometry, Pure and Applied Mathematics, 16, Marcel Dekker, Inc., New York, 1973.  Google Scholar

show all references

References:
[1]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[2]

H. Bremer, Dynamik und Regelung mechanischer Systeme, Leitfäden der angewandten Mathematik und Mechanik, 67, Vieweg+Teubner Verlag, Weisbaden, 1988. doi: 10.1007/978-3-663-05674-4.  Google Scholar

[3]

H. Bremer, Elastic Multibody Dynamics. A Direct Ritz Approach, Intelligent Systems, Control and Automation: Science and Engineering, 35, Springer, New York, 2008. doi: 10.1007/978-1-4020-8680-9.  Google Scholar

[4]

É. Cartan, Leçons sur les Invariants Intégraux, Hermann, Paris, 1971.  Google Scholar

[5]

S. R. EugsterG. Capobianco and T. Winandy, Geometric description of time-dependent finite-dimensional mechanical systems, Math. Mech. Solids, 25 (2020), 2050-2075.  doi: 10.1177/1081286520918900.  Google Scholar

[6]

H. Goldstein, Classical Mechanics, Addison-Wesley Series in Physics, Addison-Wesley Publishing Co., Reading, Mass., 1980.  Google Scholar

[7]

G. Hamel, Die Lagrange-Eulerschen Gleichungen der Mechanik, Zeitschrift für Mathematik und Physik, 50 (1904), 1-57.   Google Scholar

[8]

G. Hamel, Theoretische Mechanik, Grundlehren der Mathematischen Wissenschaften, 57, Springer-Verlag, Berlin-New York, 1978.  Google Scholar

[9]

G. Hamel, Über die virtuellen Verschiebungen in der Mechanik, Math. Ann., 59 (1904), 416-434.  doi: 10.1007/BF01445152.  Google Scholar

[10]

W. R. Hamilton, Ⅶ. Second essay on a general method in dynamics, Philos. Transac. Roy. Soc. London, 125 (1835), 95-144.  doi: 10.1098/rstl.1835.0009.  Google Scholar

[11]

R. Hermann, Differential form methods in the theory of variational systems and Lagrangian field theories, Acta Appl. Math., 12 (1988), 35-78.  doi: 10.1007/BF00047568.  Google Scholar

[12]

J. Jost, Riemannian Geometry and Geometric Analysis, Universitext, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-77341-2.  Google Scholar

[13]

J.-L. Lagrange, Théorie de la libration de la lune, Nouv. Mem. Acad. R. Sci. Bruxelles, (1780). Google Scholar

[14]

C. Lánczos, The Variational Principles of Mechanics, Mathematical Expositions, 4, University of Toronto Press, Toronto, Ont., 1949.  Google Scholar

[15]

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 2, The Classical Theory of Fields, Pergamon Press, Oxford-New York-Toronto, Ont., 1975.  Google Scholar

[16]

L. D. Landau and E. M. Lifshitz, Mechanics. Course of Theoretical Physics, Vol. 1, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960.  Google Scholar

[17]

J. M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, 218, Springer, New York, 2013. doi: 10.1007/978-1-4419-9982-5.  Google Scholar

[18]

J. M. Lee, Manifolds and Differential Geometry, Graduate Studies in Mathematics, 107, American Mathematical Society, Providence, RI, 2009. doi: 10.1090/gsm/107.  Google Scholar

[19]

O. Loos, Analytische Mechanik, Seminarausarbeitung, Institut für Mathematik, Universität Innsbruck, 1982. Google Scholar

[20]

O. Loos, Automorphism groups of classical mechanical systems, Monatsh. Math., 100 (1985), 277-292.  doi: 10.1007/BF01339229.  Google Scholar

[21]

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

[22]

J.-M. Souriau, Structure des Systèmes Dynamiques, Dunod, Paris, 1970.  Google Scholar

[23]

J.-M. Souriau, Structure of Dynamical Systems, Progress in Mathematics, 149, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-1-4612-0281-3.  Google Scholar

[24]

M. Spivak, A Comprehensive Introduction to Differential Geometry. Vol. I, Publish or Perish, Inc., Wilmington, Del., 1979.  Google Scholar

[25]

J. L. Synge, Classical Dynamics, Handbuch der Physik, Bd. III/1, Springer, Berlin, 1960, 1–225.  Google Scholar

[26]

T. Winandy, Dynamics of Finite-Dimensional Mechanical Systems, Ph.D thesis, University of Stuttgart, 2019. Google Scholar

[27]

K. Yano and S. Ishihara, Tangent and Cotangent Bundles: Differential Geometry, Pure and Applied Mathematics, 16, Marcel Dekker, Inc., New York, 1973.  Google Scholar

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