Abstract
The Lagrangian description of mechanics allows to derive the equations of motion from a variational principle based on conserved quantities of the system. In the first part of this chapter, the Lagrangian formulation of dynamics and the properties of the Lagrangian operator are synthetically reviewed starting from Hamilton’s Principle of First Action. In the second part of the chapter, the important link between continuous symmetries of the Lagrangian operator and conserved quantities of the system is introduced through Noether’s Theorem. The proof of the Theorem is reported both for material particles and for continuous systems such as fluids.
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References
Abraham, R., Marsden, J.: Foundations of Mechanics, vol. 36. Benjamin/Cummings Publishing Company Reading, Massachusetts (1978)
Arnold, V.: Mathematical Methods of Classical Mechanics. Springer, New York (1989)
Bühler, O.: A Brief Introduction to Classical, Statistical, and Quantum Mechanics, vol. 13. American Mathematical Soc. (2006)
Goldstein, H., Poole, C., Safko, J.: Classical Mechanics. Addison-Wesley, San Francisco (2001)
Gregory, R.D.: Classical Mechanics. Cambridge University Press, Cambridge (2006)
Greiner, W.: Classical Mechanics: Systems of Particles and Hamiltonian Dynamics. Springer Science & Business Media, Berlin (2009)
Hill, E.: Hamilton’s principle and the conservation theorems of mathematical physics. Rev. Mod. Phys. 23, 253–260 (1951)
Holm, D.: Geometric Mechanics. Imperial College Press, London (2008)
Holm, D., Schmah, T., Stoica, C.: Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions. Oxford University Press, Oxford (2009)
Kosmann-Schwarzbach, Y.: The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York (2011)
Landau, L., Lifshitz, E.: Mechanics. In: Course of Theoretical Physics, vol. 1. Butterworth-Heinemann, Oxford (1976)
Marsden, J.: Lectures on Mechanics, vol. 174. Cambridge University Press, Cambridge (1992)
Marsden, J., Ratiu, T.: Introduction to Mechanics and Symmetry. Springer, New York (2003)
Mušicki, D.: Canonical transformations and the Hamilton-Jacobi method in the field theory. Publications de L’Institut Mathematique Beograd 2(16), 21–34 (1962)
Neuenschwander, D.: Emmy Noether’s Wonderful Theorem. The John Hopkins University Press, Baltimore (2011)
Noether, E.: Invariante Variationsprobleme. Göttinger Nachrichten, pp. 235–257 (1918)
Olver, P.: Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, vol. 107. Springer, New York (1993)
Olver, P.: Equivalence, Invariants and Symmetry. Cambridge University Press, Cambridge (1995)
Taylor, J.: Classical Mechanics. University Science Books, Herndon (2005)
Tikhomirov, V.: Selected Works of AN Kolmogorov: Volume I: Mathematics and Mechanics, vol. 25. Springer Science & Business Media, Berlin (2012)
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Badin, G., Crisciani, F. (2018). Mechanics, Symmetries and Noether’s Theorem. In: Variational Formulation of Fluid and Geophysical Fluid Dynamics. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-59695-2_2
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DOI: https://doi.org/10.1007/978-3-319-59695-2_2
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