[Paper Review] Nonclassical Lagrangian Dynamics and Potential Maps

Constantin Udrişte|ArXiv.org|Jul 10, 2000

Advanced Differential Geometry Research9 references19 citations

TL;DR

This paper introduces a generalized Lorentz-Udrişte world-force law in semi-Riemannian geometry, showing that any first-order PDE system generates such a law via semi-Riemann-Lagrange structures. The key contribution is proving that solutions of these PDEs are harmonic or potential maps, and that the Lorentz-Udrişte law is equivalent to covariant Hamilton PDEs on the first-order jet bundle $J^1(T,M)$ with a polysymplectic structure.

ABSTRACT

Section 1 refines the theory of harmonic and potential maps. Section 2 defines a generalized Lorentz world-force law and shows that any PDEs system of order one generates such a law in suitable geometrical structure. In other words, the solutions of any PDEs system of order one are harmonic or potential maps, if we use semi-Riemann-Lagrange structures. Section 3 formulates open problems regarding the geometry of semi-Riemann manifolds $(J^1(T,M), S_1)$, $(J^2(T,M), S_2)$, and shows that the Lorentz-Udriste world-force law is equivalent to covariant Hamilton PDEs on $(J^1(T,M), S_1)$.

Motivation & Objective

• To extend classical Lagrangian dynamics to nonclassical settings using semi-Riemann-Lagrange structures.
• To show that any first-order PDE system can be interpreted as a Lorentz-Udrişte world-force law in an appropriate geometric framework.
• To establish the equivalence between the Lorentz-Udrişte world-force law and covariant Hamilton PDEs on the first-order jet bundle $J^1(T,M)$.
• To formulate open problems regarding the geometry of $J^1(T,M)$ and $J^2(T,M)$ with semi-Riemannian structures $S_1$ and $S_2$.

Proposed method

• Uses semi-Riemannian manifolds $(T,h)$ and $(M,g)$ to define energy densities and Lagrangians for maps $\varphi: T \to M$.
• Introduces a generalized energy density $E(\varphi)$ incorporating a distinguished tensor field $X^i_\alpha$ and a scalar function $c$, leading to potential maps as critical points.
• Defines the Lorentz-Udrişte world-force law via a system of second-order PDEs involving $F_j{}^i_\alpha$, $U^i_{\alpha\beta}$, and $c$, with skew-symmetric $\omega_{ji\alpha}$.
• Constructs a distinguished polysymplectic $(p+2)$-form $\Omega = \Omega_\alpha \otimes dt^\alpha$ on $J^1(T,M)$ using the Liouville form $\theta$ and its exterior derivative.
• Derives the covariant Hamilton PDEs system from the condition $X_H \llcorner \Omega_\alpha = dH$, where $H$ is the Hamiltonian observable.
• Establishes equivalence between the Lorentz-Udrişte law and the Hamilton PDEs system via the derived equations of motion on the jet bundle.

Experimental results

Research questions

• RQ1 Can every first-order PDE system be realized as a solution to a generalized Lorentz world-force law in a semi-Riemannian geometric setting?
• RQ2 How are harmonic and potential maps related to the solutions of such generalized world-force laws?
• RQ3 What is the precise geometric structure on $J^1(T,M)$ that makes the Lorentz-Udrişte world-force law equivalent to a covariant Hamiltonian PDE system?
• RQ4 What are the intrinsic geometric properties of the semi-Riemannian structures $S_1$ on $J^1(T,M)$ and $S_2$ on $J^2(T,M)$ that support this equivalence?
• RQ5 Under what conditions do continuous group actions on $M$ arise as extremals of a Lagrangian in this framework?

Key findings

• The solutions of any first-order PDE system $x^i_\alpha = X^i_\alpha(t,x)$ are harmonic or potential maps when equipped with a semi-Riemann-Lagrange structure.
• The Lorentz-Udrişte world-force law $\tau(\varphi)^i = g^{ij}\partial c/\partial x^j + h^{\alpha\beta}F_j{}^i_\alpha x^j_\beta + h^{\alpha\beta}U^i_{\alpha\beta}$ is equivalent to a covariant Hamilton PDEs system on $J^1(T,M)$.
• The Hamiltonian observable is given by $H = \left(\frac{1}{2}h^{\alpha\beta}g_{ij}x^i_\alpha x^j_\beta - f\right)dv_h$, where $f$ is a correction term related to the tensor field $X^i_\alpha$.
• The distinguished polysymplectic form $\Omega_\alpha = (g_{ij}dx^i \wedge \delta x^j_\alpha + \omega_{ij\alpha}dx^i \wedge dx^j + g_{ij}(D_\beta X^i_\alpha)dt^\beta \wedge dx^j) \wedge dv_h$ defines the geometric structure underlying the Hamiltonian system.
• The derived Hamilton PDEs system is $u^{\alpha i} = h^{\alpha\beta}x^i_\beta$ and $\delta u^{\alpha i}/\partial t^\alpha = g^{hi}h^{\alpha\beta}g_{jk}X^j_\beta(\nabla_h X^k_\alpha) + 2g^{hi}\omega_{jh\alpha}u^{\alpha j} + h^{\alpha\beta}D_\beta X^i_\alpha$, modulo terms canceled by $dv_h$.
• In the example of continuous group actions, the Lagrangian $L = \frac{1}{2}h^{\alpha\beta}g_{ij}(x^i_\alpha - \xi^i_\lambda A^\lambda_\alpha)(x^j_\beta - \xi^j_\mu A^\mu_\beta)\sqrt{|h|}$ yields maps that are extremals (potential maps) of the system.