Abstract
In this chapter it is shown how some of the basic equations of classical physics behave with respect to the requirements of the fundamental principles we have discussed earlier.
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Notes
- 1.
With the term Galilean group we mean the full set of transformations including the Euclidean covariance group and the Galilean boost.
- 2.
More correctly the gradient operator is a one-form, as pointed out in the appendix, but in Euclidean geometry vectors and one-forms do coincide.
- 3.
We’re using here the formalism from of Appendix B.
- 4.
When introducing the Lagrangian we used the symbols \(q_{i}\) and \(\dot{q}_{i}\) for the coordinates, which were called generalized coordinates and velocities. Obviously using \(\bar{x}^{k}\) and \(\dot{\bar{x}}^{k}\) in their place does not change anything when we consider that in that chapter we were not making any distinction between covariant and contravariant vectors, as is always possible in Euclidean geometry.
- 5.
This is because otherwise it would depend on the direction of the velocity, and thus it would change for rotations.
- 6.
Here by “scalar” we mean a Euclidean scalar and a Galilean-invariant, which is needed to guarantee the covariance with respect to the principle of Galilean relativity. In a general sense we can use this word to identify “numbers invariant with respect to a specific covariance group,” as we show in the next chapters. The variational approach will always carry this requirement, but varying the basic principle, and therefore the selection of a different covariance group, will originate different theories.
- 7.
In the sense that the field is generated by another source.
- 8.
Roughly speaking, a particle reacts to the value that the field has locally at the point of interaction.
- 9.
Just by formulating the same theory with another language cannot change its founding principles.
- 10.
Moreover, it has to be stressed that this formalism is ubiquitous in theoretical physics, therefore knowing it in advance can facilitate the task of understanding new theories expressed in an otherwise unknown language.
- 11.
We also recall that we were not considering the total Lagrangian because we were neglecting the free particle term. This explains why in that section we stated that \(V_{\mathrm {int}}\left( \varPhi \right) \) could be regarded as a sort of “self interaction” of the field with itself.
- 12.
As would happen, e.g., for the transformations \(\bar{\mathbf {E}}=\mathbf {E}\) and \(\bar{\mathbf {v}}\times \bar{\mathbf {B}}=\mathbf {v}\times \mathbf {B}\). The transformations cannot depend on the velocities of the bodies with respect to any reference system, but just on the relative velocity of the two reference systems.
- 13.
We recall that it is assumed that the charge density is a Galilean-invariant, thus \(\bar{\rho }=\rho \).
- 14.
For the last passage we recall that \(\mathbf {a}\cdot \left( \mathbf {b}\times \mathbf {c}\right) =-\mathbf {b}\cdot \left( \mathbf {a}\times \mathbf {c}\right) \).
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Vecchiato, A. (2017). Classical Physics, Fundamental Principles, and Lagrangian Approach. In: Variational Approach to Gravity Field Theories. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-51211-2_4
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DOI: https://doi.org/10.1007/978-3-319-51211-2_4
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