20
votes
Accepted
Functional approach vs jet approach to Lagrangian field theory
I do not know if it is good form for MO to cite one's own papers when answering a question, but I will take the chance. This matter is addressed in quite a bit of detail in my joint paper with Romeo ...
- 7,817
14
votes
Accepted
A variant of the Monge-Cayley-Salmon theorem?
Setting aside the assumption that $\phi$ be a polynomial mapping for the moment (however, see below for a construction of a large family of polynomial solutions), if one makes the 'nondegeneracy' ...
- 108k
7
votes
A linear representation of the group of jets at 0 under composition
I think you want to look at Faà di Bruno's formula, and the description of composition of formal power series, particularly the historical remarks.
The linear representation of this group that allows ...
- 108k
7
votes
Functional approach vs jet approach to Lagrangian field theory
This is meant as a long comment to the very good answer by Pedro Ribeiro.
There is a nice analog of the variational bicomplex in the functional framework. Namely, the space of differential forms on $M ...
- 5,824
6
votes
Short and elegant definition of the $C^1$ topology
The answer is almost yes if you mean in your post the following:
$C^1$-topology should mean the compact open $C^1$-topology and you should replace the $d$ by the mapping
$$T \colon C^1 (M,N) \...
- 2,388
6
votes
Accepted
Equivalence of two definitions of jets of smooth functions
The definition of $k$-th order jet as an equivalence class $[f]_x^k$ of a function $f\in C^\infty M$ at point $x\in M$, gives you a natural map
\begin{align}
\mathcal{j}^k\colon C^\infty M &\to \...
- 5,343
6
votes
Accepted
The bundle of symmetric affine connections as quotient of the second-order frame bundle
There is an 'identification', i.e., a way to interpret a torsion-free affine connection on $M$ as a section of $\check J^2(M,\mathbb{R}^n_0)/\mathrm{GL}_n(\mathbb{R})$ in such a way that every (smooth)...
- 108k
5
votes
Accepted
What is a Whitney Jet?
converted from comments:
On page 12 or the article mentioned, the author defines Whitney jets (look for the paragraph that starts with "More details.")
But basically, the classical jet ...
- 39k
5
votes
Accepted
What would be a good introductory reference for learning jet-bundle theory?
Two articles by A.M. Vinogradov provide a gentle introduction:
"Local symmetries and conservation laws", Acta Applicandae Mathematica volume 2, pages 21–78(1984)
"An informal ...
- 1,826
4
votes
In which sense are Euler-Lagrange PDE's on fiber bundles quasi-linear?
Let $\mathcal{E} \subset J^{2k}E$ be the submanifold (provided that this subset is a submanifold) of all $2k$-jets sitting in the zero-level set of $\rho(E(\mathscr{L}))$, the PDE submanifold. This is ...
- 21.5k
4
votes
Accepted
A very basic question about projections in formal PDE theory
Your conjecture is almost correct. But consider the extreme case when your equation has no integrability conditions, so that $\operatorname{coker}(\sigma(\rho_{q+1} P)) = 0$, whence $\rho^{(1)}_{q+1}(\...
- 21.5k
3
votes
Accepted
Splitting of higher order jet sequence
Look at the paper P. Jahnke and I. Radloff, Splitting jet sequences. They classify such splittings on compact Kaehler manifolds. Those which admit a vector bundle with splitting jet sequence are ...
- 26.3k
3
votes
Accepted
Semi-holonomic jets in synthetic differential geometry
According to Liebermann's Introduction to the theory of semi-holonomic jets p.177:
a local section $s:U\subset M\to J^1E$ is said to be adapted at $x\in U$ if $s(x)=j^1_x(\beta \circ s)$ where $\...
- 5,343
3
votes
Accepted
Does the Banach algebra of jets have the approximation property?
Is $x_0+\tau x_0' \mapsto (x_0'(\cdot), x_0(0))$ a linear homeomorphism from $J^1$ to $C([0,1])\times \mathbb R$? Wouldn't that imply that $J^1$ inherits the approximation property from $C^0([0,1])$?
- 46
2
votes
Constructing jet bundles from a cocycle of smooth transition functions
I would not really call this an answer but rather an extended comment. I think that the answer by @IgorKhavkine is correct but at the same time misleading in a certain sense. The point about this is ...
- 1,624
2
votes
Accepted
In which sense are Euler-Lagrange PDE's on fiber bundles quasi-linear?
Igor gave a coordinate independent definition of quasilinear equations which the Euler-Lagrange equations satisfy. Still missing is a definition of quasilinear differential operator, which the Euler-...
- 5,343
1
vote
Precise definition of a linear total differential operator
I think I managed to solve this, but I have not verified this construction completely and rigorously. Still, I think it works and I am posting an answer at least for documentation and a more explicit ...
- 2,792
1
vote
Equivalence of two definitions of jets of smooth functions
This can be generalized to the following. Let $E$ and $F$ be smooth vector bundles over a smooth manifold $M$. Then differential operators of order $\leqslant k$ from $E$ to $F$ form a locally free ...
- 538
1
vote
Integrability conditions for differential equations on $J^\infty$
Section 4 of the paper "Geometry of Differential Equations" by B. Kruglikov and V. Lychagin (IHES/M/07/04) states that formal integrability & analyticity are sufficient for the existence of a ...
- 1,826
1
vote
Integrability conditions for differential equations on $J^\infty$
There exists a theory of integrability for differential equations on vector bundles over infinite dimensional jet bundles or, more generally, diffities. Although several vector fields on such ...
- 493
1
vote
Accepted
Constructing jet bundles from a cocycle of smooth transition functions
You start with the trivializations $S \times U_i \to U_i$ over your cover. Let me presume that you can take for granted the construction of the jet bundles $J^r(S \times U_i) \to U_i$, whose typical ...
- 21.5k
Only top scored, non community-wiki answers of a minimum length are eligible