On a smooth manifold of dimension $n$, the application value of the canonical $1$-form, the Liouville form on $T^*(X)$, to the Hamiltonian mechanics is well known; $T^*(X)$ is a degree $1$-Jet bundle. My question is Do canonical forms similar to the Liouville form exist on higher degree Jet bundles? I ask this because, beyond the invariant sub-principal symbol of a pseudodifferential operator, nothing much seems to be known to handle multiple characteristic problems, especially of the non-involutive type. I am aware of Ivrii-type Fuchsian operators, already posing great difficulties.
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asked May 10, 2021 at 7:57
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3$\begingroup$ The cotangent bundle is not a jet bundle. Each jet bundle of order $k$ maps onto all lower order jet bundles, of all orders less than $k$. So they all map onto the $0$-jet bundle, the trivial bundle. But the cotangent bundle does not map onto the trivial bundle. $\endgroup$– Ben McKayCommented May 10, 2021 at 8:05
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1$\begingroup$ The jet bundles bear a canonical, diffeomorphism invariant, exterior differential system, often called the contact system. You can read about on p. 22 of the book Exterior Differential Systems. $\endgroup$– Ben McKayCommented May 10, 2021 at 8:10
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$\begingroup$ Thank you Prof.McKay,but honestly,I expected a different answer,namely that if you consider any 1-form on X and its ext.derivative pulled back on T*(X),then as a 1-form on $\endgroup$– Nagaraj IyengarCommented May 10, 2021 at 10:18
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$\begingroup$ Nagaraj Iyengar: your comment got cut off before you finished it. $\endgroup$– Ben McKayCommented May 10, 2021 at 10:32
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$\begingroup$ Thank you, posting my comment again: I had expected that if you consider any 1-form of $X$ and its exterior derivative pulled back on $T^*(X)$, then as a 1-form on $X$ 'associated' to the Liouville form, there must be a commutative diagram (Godbillon, C 1969). Moreover, in pseudodifferential calculus, one always works outside the zero section of $T^*(X)$, but always overcomes the 'diagonal singularity' by the 'splitting' with a properly supported operator + one with smooth kernel. Thus the question effectively asks how to construct the commutative diagram in higher Jets. $\endgroup$– Nagaraj IyengarCommented May 11, 2021 at 2:26
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1 Answer
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The $k$-jet bundle $J^k$ of $k$-jets of real valued functions on a manifold $M$ has an obvious map $J^k\to J^1$, if $k\ge 1$, smooth and diffeomorphism invariant, taking the $k$-jet of a function to its $1$-jet. The $1$-jet bundle has an obvious splitting $J^1=J^0 \oplus T^* X$, mapping each $1$-jet to its $0$-jet and exterior derivative of any function, as calculated from its $1$-jet. Pull back the Liouville form $\lambda$ from $T^*X$ to $J^1$, and to $J^k$, to obtain a $1$-form $\lambda$ on $J^k$ so that, for any $C^{k+1}$ function $f$, its $k$-jet $j^k f$, as a section of $J^k$, satisfies $(j^k f)^*\lambda=df$. Is that what you are looking for?
answered May 11, 2021 at 9:34
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$\begingroup$ Thanks again Prof. McKay. I request some clarifications: 1. Does your global construction take note of the 'obstruction' to the global definability via non-degenerate phase functions of a Lagrangian submanifold? (please refer Lees, J.A, Trans Am, Vol 250, pp 213-222 (1979)) 2. At the level of the Hamiltonian and via Egorov conjugation relative to a single fiber coordinate, how to compute the base diffeo whose 'associated' canonical map completes the closed diagram for vector fields via pull backs? $\endgroup$ Commented May 20, 2021 at 3:03
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1$\begingroup$ I am sorry that I don't understand either question. The construction doesn't require a Lagrangian manifold, so it doesn't require that any obstruction vanish. I don't know what Egorov conjugation, the base diffeo, the canonical map, the closed diagram, or the pullbacks are; these terms do not appear in the paper you cite. $\endgroup$ Commented May 20, 2021 at 16:24