The Use of the Linear Combination of Physical Solutions in the Study of the Nature of the Stationary Point of Action
DOI:
https://doi.org/10.14393/BEJOM-v1-n1-2020-48000Keywords:
Classical Mechanics, Action, Hamilton’s Principle, Functional CalculusAbstract
Hamilton’s principle states that among all curves between two end points, the path actually followed by a physical system will always be the one that gives an stationary value (minimum, maximum or a saddle points) to the Hamilton’s action (an integral over time of the difference between the kinetic and the potential energy, taken between the initial time and the final time of the development of the system). It is common to use a mathematical tool called Calculus of Variations to study the Hamilton’s principle. The Calculus of Variations deals with functionals (functions of functions) and is a more general and more complex version of the usual calculus that we learn in university. In this paper we present an alternative, and simpler, approach to the study of the Hamilton’s principle. We study the nature of the action’s stationary space-time trajectory of three physical systems: free-particle, vertical launch and harmonic oscillator, using as virtual motion a linear combination of the physical solution of this three systems. We find evidences that the physical solution of the free-particle problem leads to a minimum on its action. The same results occours on the vertical launch problem. The physical solution of the harmonic oscillator leads to a minimum or a saddle point on its action deppending on the time interval of the development of the system.
Downloads
References
LEMOS, N .A. Mecânica Analítica, São Paulo, Editora Livraria da Física, 2007.
THORNTON, S. T.; MARION, J. B. Dinâmica Clássica de Partículas e Sistemas, São Paulo, Editora Cengage Learning, 2011.
GRAY, C. G.; TAYLOR, E. F. When action is not least, American Journal of Physics, v. 75, n. 5, p. 434-458, 2007.
FREIE, W. H. C., A derivada funcional de segunda ordem da ação: investigando minimalidade, maximalidade e ponto de sela, Revista Brasileira de Ensino de Física, v.34, n. 1, 1301, 2012.
MORICONI, M. Condition for minimal harmonic oscillator action,American Journal of Physics, v. 85, p. 633-634, 2017.
STEWART, J. Cálculo - Vol. 2, 6ª edição, São Paulo, Editora Pioneira Thomson Learning, 2009.
Downloads
Published
Issue
Section
License
Copyright (c) 2020 Fábio Pascoal dos Reis, Uilian de Oliveira Pereira, Pablo Henrique Menezes, Elisângela Aparecida Y. Castro

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
- Articles published from 2025 onwards are licensed under the CC BY 4.0 license. By submitting material for publication, authors automatically agree to the journal’s editorial guidelines and affirm that the text has been properly reviewed. Simultaneous submission of articles to other journals is prohibited, as is the translation of articles published in this journal into another language without proper authorization.
- Articles published prior to 2025 are licensed under the CC BY-NC 4.0 license.