Recently, I've been wondering how to rewrite the standard Euler-Lagrange equations: \begin{align} \dfrac{\partial L}{\partial q^i} - \dfrac{d}{dt} \left(\dfrac{\partial L}{\partial \dot{q}^i}\right) &= 0 \end{align} (all derivatives being evaluated at appropriate points along a stationary curve of the action functional) without referencing the coordinates $(q^1, \dots q^n, \dot{q}^1, \dots, \dot{q}^n)$ on the tangent bundle. So, I tried to rewrite things as much as possible using only "natural operations" on the tangent bundle, like Lie-derivative, exterior derivative etc. Of course, I didn't succeed (and any articles I tried to search online were too abstract to understand), so what I did was expand the total time derivative: \begin{align} \dfrac{\partial L}{\partial q^i} - \left[\dfrac{\partial^2 L}{\partial q^j\partial \dot{q}^i}\dot{q}^j + \dfrac{\partial^2 L}{\partial \dot{q}^j\partial \dot{q}^i}\ddot{q}^j \right] &= 0. \end{align} This gave me the idea that perhaps thinking of the Lagrangian, $L$, as a function on the tangent bundle $TQ \to \Bbb{R}$ is perhaps not the most appropriate/natural setting. It seems that the tangent bundle only has coordinates $(q, \dot{q})$, whereas the Euler-Lagrange equations which are second order differential equations, involving $\ddot{q}$. So, it seemed to me that it would be nice to construct a new manifold $M$, from the original "configuration manifold" $Q$, so that on the new manifold $M$, we have local coordinates $(q^i, \dot{q}^i, \ddot{q}^i)$, for $1 \leq i \leq n$.

To formalize this idea of introducing a larger manifold with extra coordinates for higher derivatives, my plan was to mimic the construction of $TQ$ as much as possible.

Let $k \geq 1$ be an integer and let $Q$ be a smooth manifold modeled on a Banach space $E$. Now, we define a relation $\sim_k$ on the set of all smooth curves $\gamma:I \to Q$, where $I$ is an open set in $\Bbb{R}$ containing $0$, by declaring $\gamma_1 \sim_k \gamma_2$ if and only if there is a chart $(U, \alpha)$ of $Q$ such that (the composition below makes sense and) for every $r \in \{0, \dots, k\}$, we have \begin{align} (\alpha \circ \gamma_1)^{(r)}(0) &= (\alpha \circ \gamma_2)^{(r)}(0). \end{align} A tedious, but straight forward induction exercise and chain rule shows that this relation doesn't depend on the choice of chart $(U, \alpha)$, so we're actually justified with using the notation $\sim_k$ without referencing the chart. This is also an equivalence relation. For the lack of a better name, I shall denote the quotient set of equivalence classes as \begin{align} C^kQ := (\text{smooth curves in $Q$})/\sim_k, \end{align} and I shall call it "the $k^{th}$ order contact manifold of curves in $Q$". Given a smooth curve $\gamma$, we shall indicate the equivalence class either as $C^k\gamma$ or $[\gamma]$, whichever is more convenient. Next, I outline how I put a manifold structure.

I realized that this space has quite a bit of structure: we can define a projection map $\pi_k : C^kQ \to Q$ by sending $C^k \gamma \mapsto \gamma(0)$. We can even make this into a smooth manifold as follows: given a chart $(U, \alpha)$ on $Q$, we define the chart $(C^kU, C^k \alpha)$ by defining $C^kU := \pi_k^{-1}[U]$ and $C^k \alpha : C^kU \to \alpha[U] \times E^k$, \begin{align} C^k \alpha([\gamma]) &:= \left( (\alpha \circ \gamma)(0), (\alpha \circ \gamma)'(0), \dots (\alpha \circ \gamma)^{k}(0) \right) \end{align} This is a well-defined map because of how the equivalence relation was defined. It is also easily seen to be a bijective map, and also, if $(V, \beta)$ is another chart on $Q$ with $U \cap V \neq \emptyset$, then using the chain rule, it is straightforward (though tedious) to seem that $(C^k \beta) \circ (C^k \alpha)^{-1}$ is a smooth map between open subsets of Banach spaces.

Also, I only just recently read up about fiber bundles, but I believe that based on what I've constructed, we have that $\pi_k :C^kQ \to Q$ is a fiber bundle, with typical fiber $E^k$, whereby the maps $C^k\alpha$ defined above provide us with the local trivializations. Is this the correct way of using the terminology?

Here are my questions:

  • Have people considered such spaces $C^kQ$? Are they interesting manifolds to study, and am I right in thinking that these manifolds would be a more appropriate setting to formulate Lagrangian mechanics? I believe that this is true, because if for some reason we decided to consider a Lagrangian which depends on higher derivatives of the curve, then the tangent bundle alone is insufficient to capture such information. In any case, I would appreciate some confirmation/denial along with references (if any).

  • Is it possible to formulate the Euler-Lagrange equations in a coordinate-free manner, perhaps using such spaces? If yes, I'd appreciate some references (which hopefully aren't too abstract).

  • For $k=1$, this construction yields precisely the tangent bundle, in which case, the fiber over each point $x \in Q$, namely $T_xQ$ can be given a natural vector space structure. However, for $k>1$, am I right in saying that we cannot endow each fiber with a vector space structure? Of course, if we choose a particular chart $(U, \alpha)$, we can establish a bijection between the fiber $\pi_k^{-1}(\{x\})$ and $E^k$, and hence inherit a vector space structure that way, but I believe that this is not a chart-independent construction. Is this right?

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    $\begingroup$ I didn't read everything you wrote, but the idea of taking the $k$-th derivative tangent vectors is described by looking at the space of $k$-jets $\Bbb R\to M$ at $0$. The language of jets and jet bundles is natural when tracking higher derivatives than $1$. However higher derivatives are messy, and this language is (imo) unpleasant. You can find a short introduction in these notes by Ballmann or in this book in chapter 4. $\endgroup$ – s.harp Jun 1 '20 at 9:20
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    $\begingroup$ @s.harp I see, so what I've been calling $C^kQ$ is more commonly referred to as the $k^{th}$ jet space $J^k(\Bbb{R}, Q)$. Thanks for pointing me in the right direction regarding some "buzz words" to search, and for the references. $\endgroup$ – peek-a-boo Jun 1 '20 at 9:28

Regarding your second question, see Equation 8 in Crampin, M. (1981). On the differential geometry of the Euler-Lagrange equations, and the inverse problem of Lagrangian dynamics. Journal of Physics A: Mathematical and General, 14(10), 2567–2575. https://doi.org/10.1088/0305-4470/14/10/012


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