Abstract
One of the powerful features of the variational approach is its generality. It is in fact based on a simple principle, stating that physical laws share the common characteristics of minimizing the action of a system, and as we have seen this can be applied to mechanics (i.e., the dynamics of particles) as well as to the dynamics of fields.
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Notes
- 1.
Given the scope of this book, the exposition is limited to classical physics. However, as long as the mechanism of the so-called quantization is understood, the same techniques can be applied to quantum models.
- 2.
And also five common notions, which can be regarded as other axioms.
- 3.
The term “rules of correspondence” was first used in this sense by the philosopher Rudolf Carnap (1891–1970) in his concept of a scientific theory as an axiomatic formal system. In the case under discussion, strictly speaking, the definitions tell us which are the “characters of the game.” Their correspondence with entities of the real world, however, is implicit in its use. For Euclid’s Elements these regard the constructions that could be done with rulers and compasses, whereas in other works using these geometric theories as its fundamental tool this “mapping” could concern different objects.
- 4.
It has to be stressed that this connection must not be interpreted as a one-to-one identification between physics and geometry. For example, a particle in classical physics can correspond to a geometric point, but in a gravity theory this model includes a property called mass, and in electrodynamics we have an electric charge.
- 5.
There are many such theories, and their predictions for these particles can vary considerably.
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Vecchiato, A. (2017). The Geometrical Character of Physics Theories. In: Variational Approach to Gravity Field Theories. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-51211-2_2
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DOI: https://doi.org/10.1007/978-3-319-51211-2_2
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