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The Classical Formulations of Dynamics of Hamilton and Lagrange

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Geometry from Dynamics, Classical and Quantum

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Abstract

The present chapter is perhaps the place where our discourse meets more neatly the classic textbooks on the subject. Most classical books concentrate on the description of the formalisms developed by Lagrange and Euler on one side, and Hamilton and Jacobi on the other and commonly called today the Lagrangian and the Hamiltonian formalism respectively. The approach taken by many authors is that of postulating that the equations of dynamics are derived from variational principles (a route whose historical episodes are plenty of lights and shadows [Ma84]).

On ne trouvera point de Figures dans set Ouvrage. Les méthodes que j’y expose ne demandent ni constructions, ni raisonnements géométriqus ou méchaniques, mais seulement des opérations algébriques, assujetties à une march réguliere et uniforme.

Joseph-Louis Lagrange, Mécanique Analytique, Avertissement de la premiére édition, 1788.

‘The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings, but only algebraic operations, subject to a regular and uniform rule of procedure’.

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Notes

  1. 1.

    The same definitions hold for an infinite-dimensional symplectic linear space without the dimension relations.

  2. 2.

    The argument will work unchanged in the infinite-dimensional instance applying Zorn’s Lemma.

  3. 3.

    Notice that switch from ‘left action’ to ‘right invariance’ in the previous statement. It happens because of \(TR_g \xi _G(h) = \frac{d}{ds} R_g((\exp s\xi ) h) \mid _{s = 0} = \frac{d}{ds} \exp s\xi (hg) = \xi _G(hg)\).

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Authors and Affiliations

  1. Departamento de Física Teórica, Universidad de Zaragoza, Zaragoza, Spain

    José F. Cariñena

  2. Departamento de Matemáticas, Universidad Carlos III de Madrid, Madrid, Spain

    Alberto Ibort

  3. Dipartimento di Fisiche, Universitá di Napoli “Federico II”, Napoli, Italy

    Giuseppe Marmo

  4. INFN Sezione di Bologna, Universitá di Bologna, Bologna, Italy

    Giuseppe Morandi

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  1. José F. Cariñena
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  2. Alberto Ibort
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  3. Giuseppe Marmo
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  4. Giuseppe Morandi
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Correspondence to José F. Cariñena .

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Cariñena, J.F., Ibort, A., Marmo, G., Morandi, G. (2015). The Classical Formulations of Dynamics of Hamilton and Lagrange. In: Geometry from Dynamics, Classical and Quantum. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9220-2_5

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  • DOI: https://doi.org/10.1007/978-94-017-9220-2_5

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