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• W1690703721 abstract "This paper combines some of our ideas to obtain the multitime variants of Hamilton-Jacobi theory. Section 1 introduces some Riemannian first order jet bundles. Section 2 describes the multitime Hamilton-Jacobi PDE system connected to the path independent curvilinear integral action. Section 3 analyzes the multitime HamiltonJacobi divergence PDE connected to multiple integral action. Section 4 studies the harmonic map Lagrangian. Section 5 underlines the novelty of our results. Key–Words: Hamilton-Jacobi PDE, path independent curvilinear integral action, multiple integral action, submanifolds. 1 Riemannian first order jet bundles Let (T, h) be a Riemannian manifold with m dimensions, and (M, g) be a Riemannian manifold with n dimensions. Then ( J1(T, M), h + g + h−1 ∗ g) is a Riemannian first order jet bundle with m + n + mn dimensions and ( J(T ×M,T ), h + g + h + h−1 ∗ h + h−1 ∗ g ) is a Riemannian first order jet bundle with n + 2m + mn + m2 dimensions. Particularly, the Riemannian manifolds (Rm, δαβ), (Rn, δij) determine the Riemannian first order jet bundles J1(Rm, Rn), J1(Rm ×Rn, Rm). 2 Multitime Hamilton-Jacobi PDE system In mathematics, the Hamilton-Jacobi PDE is a necessary condition describing extremal geometry in generalizations of calculus-of-variations problems. In physics, the Hamilton-Jacobi PDE is equivalent to Newton Law of Motion, Lagrangian Mechanics and Hamiltonian Mechanics. The Hamilton-Jacobi PDE is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely. We improve this point of view adding the multitime version of the Hamilton-Jacobi PDE (see [3]-[11]). Let t = (tα) ∈ Rm and x = (xi) ∈ Rn. Let S : Rm × Rn → R be a C1 function to whom we attach the constant level sets Σc : S(t, x) = c. Suppose Σc are submanifolds in Rn+m = Rm × Rn, i.e., the normal vector field ( ∂S ∂tβ , ∂S ∂xi ) is not zero anywhere. Let Γ : (t, x(t)) be an m-sheet transversal to the submanifolds Σc. Then the function c(t) = S(t, x(t)) has nonzero partial derivatives ∂c ∂tα (t) = ∂S ∂tα (t, x(t)) + ∂S ∂xi (t, x(t)) ∂xi ∂tα (t) = ∆α(t, x(t), xγ(t)) 6= 0. We accept Lα(t, x(t), xγ(t)) = ∆α(t, x(t), xγ(t)), i.e., the Lagrange 1-form Lα is just the total derivative of the function S(t, x(t)). It follows pγαi = ∂Lα ∂xi γ = ∂∆α ∂xγ = ∂S ∂xi δ α" @default.
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• W1690703721 date "2009-08-20" @default.
• W1690703721 modified "2023-10-15" @default.
• W1690703721 title "Multitime Hamilton-Jacobi theory" @default.
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