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Lagrangian mechanics on the bundle

It is part of the modern method of physics that all classical objects eventually get the fiber bundle treatment, and Lagrangian mechanics is indeed one of them. While this is somewhat simple to do with classical mechanics for point particles, things get a lot harder once we start involving fields, gauge theories and instantons.

From a bundle point of view, the field configurations correspond to some sections of a vector bundle $Y$ over our spacetime $M$, with projection $\pi : Y \to M$. The configuration space is then the set of smooth sections of $E$, noted $\Gamma^\infty(M, Y)$. The action is then just a linear functional of the form

\begin{eqnarray} S : \Gamma^\infty(M, Y) \times \mathscr{T} &\to& \mathbb{R}\\ (\phi, U) &\mapsto& S(\phi|_{U}, U) \end{eqnarray}

where $\mathscr{T}$ is the set of open sets of $M$. $S$ is the action over $U$ of the field $\phi$, and it is linear in its first argument.

From here, we need to find out what the Lagrangian is. The Lagrangian itself has to be a function of $\phi$, therefore map $E$ in some way to $\Lambda^n U$, the set of $n$-forms on $U$, so that we can define the usual manifold integral

\begin{eqnarray} S(\phi|_{U}, U) = \int_U \mathcal{L} \end{eqnarray}

We could simply put $\mathcal{L}$, the lagrangian density, as a function of $\phi$ directly, but this would not be much in line with the way it is usually defined, where its arguments are the values of the section $\phi$ at individual points, as well as its derivatives. To do this, we need to introduce the notion of the jet bundle.

1. Reminder on fiber bundles

First, a brief reminder. We're going to use heavily the mapping of our quantities to coordinates, so it will be useful to remind us of how to do this.

The coordinates of the manifold, in a coordinate neighbourhood $U$, has the chart $\phi : U \to \mathbb{R}^n$, noted $(U, \phi)$, such that

\begin{eqnarray} \phi(p) = (x_1, \ldots, x_n) \end{eqnarray}

We will generally note this as $x(p)$ or $x$. The switch to another coordinate system $(U', \phi')$ is done via the transition map $\phi' \circ \phi^{-1}$, so that

\begin{eqnarray} \phi'(p) = \phi'(\phi^{-1}(x_1, \ldots, x_n)) \end{eqnarray}

If we denote our new coordinates $\phi'(p) = y(p)$, we will usually denote that transition informally, if there are no ambiguity as to its meaning, by $\phi' \circ \phi^{-1}(x) = y(x)$, so that $y(x(p)) = y(p)$.

The fiber bundle also has a local trivialization of the form

\begin{eqnarray} \psi : \pi^{-1}(U) \subset Y &\to& U \times F \end{eqnarray}

with $F$ the typical fiber. Therefore, if we combine the local trivialization with the charts of $U$ and $F$, $\phi_U$ and $\phi_F$, we obtain that there is a coordinate system of $Y$ such that

\begin{eqnarray} \forall y \in Y, \phi_Y(y) = (x_1, \ldots, x_n, y_1, \ldots, y_m) \end{eqnarray}

As long as the changes of the charts of $U$ and $F$ are independent, the coordinates will vary independently (the change in the trivialization will not change this as the local trivialization respects the fiber). But as a counterexample, consider the case of the tangent bundle :

The jet bundle

Jet manifolds

The $k$-jet space

Jet bundles are defined on another fiber bundle. In our case, the bundle

\begin{eqnarray} \pi : E \to M \end{eqnarray}

Linear functionals on the jet space

The action of a field theory is typically written as the integral

\begin{eqnarray} S &=& \int_U L(\phi, d\phi) \end{eqnarray}

We can recognize here that $L$ is a function mapping from the jet bundle of our scalar field to the space of $n$-forms (to make it possible to integrate it over).

In the language of jets, we say that our Lagrangian maps the jet of a section to $n$-forms over $U$. More precisely, assuming a measure $n$-form $\omega$ over $U$, our action for a section $\phi$ is

\begin{eqnarray} S[\phi] &=& \int_U L(j_* \phi) \omega \end{eqnarray}

While we generalize to jets of any degree, the Lagrangian will typically stop at the first, or at most second, jet, but there is little harm in keeping things general for now.

Global Lagrangian

So far, our description of the Lagrangian has been mostly local.

Bibliography

1. G. Sardanashvily, Five lectures on the jet manifold methods in field theory
2. U. Schreiber, Higher prequantum geometry
3. P. McCloud, Jet Bundles in Quantum Field Theory: The BRST-BV method
4. G. Barnich, Introduction to Classical Gauge Field Theory and to Batalin-Vilkovisky Quantization
5. M. Gotay, A Multisymplectic Framework for Classical Field Theory and the Calculus of Variations, 1991

Last updated : 2023-04-21 13:08:04