Abstract
In this chapter, we develop the Lagrangian and Hamiltonian formulations of continuous systems, which serves as an introduction to the analysis of fields. This chapter is optional in the sense that it is dependent on the previous chapters, but will not be necessary for the following chapter.
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Notes
- 1.
For this and all subsequent variations, we ignore all 2nd-order and higher terms in \(\delta y\) and its partial derivatives.
- 2.
To explicitly denote the x and t dependence of the functional derivative, one should write \(\delta S/\delta y(x, t)\) instead of the shorthand notation \(\delta S/\delta y\).
- 3.
Note that the action integral is defined with respect to the 3-dimensional coordinate volume element \(\mathrm d^3 x\) and not with respect to the 3-dimensional invariant volume element \(\mathrm dV\), which differs from \(\mathrm d^3 x\) by a factor of \(\sqrt{\det g}\) (See (A.61) in Appendix A.5). The above definition is consistent with the interpretation of \({\mathscr {L}}\) being a density with respect to coordinate transformation. If you don’t want to worry about such a distinction, just always work in Cartesian coordinates \(x^i=(x,y, z)\) for which \(\mathrm d^3 x\) and \(\mathrm dV\) are numerically equal to one another.
- 4.
In Chap. 11, we will introduce similar notation to describe the components of four-dimensional vectors in the context of special relativity. In that context, there is an implied geometric structure to the spacetime, which is Minkowskian; but we can get by without introducing that formalism here.
- 5.
For the field y(x, t), we will often use \(\dot{y}\) to denote the partial derivative \(y_{, t} \equiv \partial y/\partial t\).
- 6.
Such a symmetry is sometimes called a divergence symmetry, as opposed to a variational symmetry, which would have \(\delta {\mathscr {L}}=0\).
- 7.
For all these examples, we will work in Cartesian coordinates \(x^i=(x,y, z)\) for which \(\sqrt{\det g}=1\) and the Lagrangian densities for the 3-dimensional Klein-Gordon field have the explicit forms given in (10.121) and (10.137). If, instead, you would like to do a particular calculation in e.g., spherical coordinates \((r,\theta ,\phi )\), then you should first multiply the Lagrangian densities given in (10.121) and (10.137) by \(\sqrt{\det g} = r^2\sin \theta \) before assessing the form of the symmetry and calculating the conserved current \(J^\alpha \), etc.
- 8.
It turns out that this definition of Poisson brackets is independent of the choice of time t. See e.g., Torre (2016) for a proof.
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Benacquista, M.J., Romano, J.D. (2018). Lagrangian and Hamiltonian Formulations for Continuous Systems and Fields. In: Classical Mechanics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-68780-3_10
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DOI: https://doi.org/10.1007/978-3-319-68780-3_10
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