June  2021, 13(2): 167-193. 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  June 2021 Early access  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, 2021, 13 (2) : 167-193. 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.

[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.

[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.

[4]

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

[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.

[6]

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

[7]

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

[8]

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

[9]

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

[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.

[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.

[12]

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

[13]

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

[14]

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

[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.

[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.

[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.

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[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.

[24]

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

[25]

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

[26]

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

[27]

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

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.

[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.

[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.

[4]

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

[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.

[6]

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

[7]

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

[8]

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

[9]

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

[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.

[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.

[12]

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

[13]

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

[14]

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

[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.

[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.

[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.

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[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.

[24]

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

[25]

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

[26]

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

[27]

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

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