Abstract
At the age of 19, recognized for his extraordinary mathematical abilities, Joseph Louis de Lagrange (1736–1813) was appointed professor of geometry and mechanics at the Royal Artillery School at Turin, Italy, his birthplace. Here he developed his method of variations, invented earlier by Euler (in 1744) who later named it the calculus of variations. Lagrange left Turin in 1766 to become director of the Berlin Academy of Sciences until 1787 when, at the invitation of King Louis XVI of France, he was appointed to the Paris Academy of Sciences.†† Shortly thereafter his most celebrated work, Mécanique Analytique, appeared in 1788, nearly a century after the appearance of Newton’s Principia. Therein, Lagrange sets down an energy based approach for dynamics—the analysis of motion.
Lagrange’s life and times are sketched in the translators’ “Introduction” in Lagrange’s Analytical Mechanics cited in the chapter References. See also Truesdell’s Essays.
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References
GREENWOOD, D. T., Principles of Dynamics, Prentice-Hall, Englewood Cliffs, New Jersey, 1965. Cast at an intermediate level comparable to the present text, this book provides an excellent resource for collateral study. Lagrange’s equations, including a thorough discussion of constraints and the method of virtual work, are discussed in Chapter 6. The theory of vibrations, including discussion of degenerate systems, is treated in Chapter 9.
HOUSNER, G. W., and HUDSON, D. E., Applied Mechanics: Dynamics, 2nd Edition, Van-Nostrand, Princeton, New Jersey, 1959. Lagrange’s equations applied to thetheory of small, free, and forced vibrations are studied in Chapter 9. The authors provide good preparation throughout for the student’s subsequent study of advanced topics in dynamics. Some examples and problems in the current chapter are modeled after those presented in this book.
LAGRANGE, J. L., Analytical Mechanics, translated from the Mécanique analytique novelle édition of 1811, editors A. Boissonnade and V. V. Vagliente, Kluwer, Dordrecht, The Netherlands, 1997. The original of this enduring classical work often is unavailable for study, so this English translation is a most helpful substitute. The translators’ “Introduction” provides an interesting sketch of the life and times of Lagrange. Lagrange’s treatise, based on the method of virtual work, begins with principles of statics in Part I; the greater emphasis is on the various principles of dynamics in Part II.
LANCZOS, C, The Variational Principles of Mechanics, University of Toronto Press, Toronto, Canada, 1949. A beautifully written text highly recommended for advanced study. The focus, however, is mainly on the theoretical aspects of Lagrange’s differential equations of motion and the Hamilton-Jacobi canonical theory of equations. There are very few, though well chosen examples throughout. The method of undetermined multipliers is applied in the formulation of Lagrange’s equations for systems with constraints. Our discussion of nonholonomic constraints mirrors that presented in this text.
LONG, R. R., Engineering Science Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey, 1963. Vector and Cartesian tensor methods are integrated in this treatment of topics on engineering mechanics. Hamilton’s principle and Lagrange’s equations are introduced in Chapter 3. The latter, however, are not uniformly applied throughout the text, which otherwise provides a good resource for collateral reading and for additional problems and examples.
MARION, J. B., Classical Dynamics of Particles and Rigid Bodies, Academic, New York, 1965. In Chapter 9, Lagrange’s equations are derived from Hamilton’s principle, and the method of undetermined multipliers to characterize constraints on a system is introduced. Multidegree of freedom systems are studied in Chapter 14. Applications of Euler’s equations and the energy principle for rigid bodies, including the spinning top problem, are studied in Chapter 13. See also MARION, J. B., and THORNTON, S. T., Classical Dynamics of Particles and Rigid Bodies, Harcourt Brace, New York, 1995.
PARS, L. A., A Treatise on Analytical Dynamics, Ox Bow Press, Woodbridge, Connecticut, 1965. This text provides many examples for advanced readers.
ROSENBERG, R. M., Analytical Dynamics of Discrete Systems, Plenum, New York, 1977. This is a carefully written, thorough treatment of analytical dynamics based on the geometry of the configuration space of generalized coordinates, including a precise presentation of the geometry of constraints and virtual displacements, and their relation to constrained systems. Chapter 9 is a clear analysis and description of D’Alembert’s principle that leads in subsequent chapters to Lagrange’s equations for arbitrary systems. There are many worked examples throughout.
SYNGE, J. L., and GRIFFITH, B. A., Principles of Mechanics, 3rd Edition, McGraw-Hill, New York, 1959. Chapter 14 treats the general problem of the spinning top and the gyroscope by Euler’s equations and the energy method, and Chapter 15 deals with Lagrange’s equations with several examples, including the top problem.
TRUESDELL, C, Essays in the History of Mechanics, Springer-Verlag, Berlin, Heidelberg, New York, 1968. The author argues in Chapters 2 and 5 that Euler’s laws are more general than Lagrange’s equations, pointing specifically to the significance of Euler’s principle of moment of momentum and noting that it is never mentioned or used by Lagrange.
WHITTAKER, E. T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 3rd Edition, Cambridge University Press, Cambridge, 1927. This classical volume, highly recommended to advanced readers, remains one of the best mathematical treatments of analytical dynamics. The treatise focuses entirely on the Lagrangian and Hamiltonian theories and includes many worked examples throughout. The reader may find the problems rather challenging, however. The theory of vibrations, including degenerate systems, is analyzed thoroughly in Chapter VII; nonholonomic systems are studied in Chapter VIII, and extension of Hamilton’s principle to conservative and nonconservative, nonholonomic systems follows in Chapter IX.
YEH, H., and ABRAMS, J. I., Principles of Mechanics of Solids and Fluids, Vol. 1, Particle and Rigid Body Mechanics McGraw-Hill, New York, 1960. Lagrange’s equations and some examples, including application to the vibrations of a structure, are discussed in Chapters 13.
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Beatty, M.F. (2006). Introduction to Advanced Dynamics. In: Principles of Engineering Mechanics. Mathematical Concepts and Methods in Science and Engineering, vol 33. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-31255-2_7
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