Euler-Lagrange-Hamilton dynamics with fractional action

Abstract

Abstract: Our aim is three-fold: to point out that the fractional integral actions are coming from Stieltjes actions, to find the roots and the geometry of some Euler-Lagrange or Hamilton ODEs or PDEs, to evidentiate some ideas that include the fractal theory of solids. Section 1 discusses the Euler-Lagrange ODEs associated to single-time Stieltjes actions. Teir dual Hamilton ODEs are analized in Section 2. Section 3 studies the geometry associated to single-time Euler-Lagrange or Hamilton operators. Section 4 analyzes the Euler-Lagrange PDEs associated to multitime Stieltjes actions (multiple or curvilinear integrals). Section 5 formulates the multitime perimetric problem of non-renewable resources. Section 6 studies the Hamilton PDEs associated to multitime Stieltjes actions. Section 7 describes the geometry associated to multitime Euler-Lagrange or Hamilton operators (dynamical connection and semi-spray, Poincaré-Cartan form, Hamilton-Poisson systems on jet bundle). Section 8 formulates a multitime Hamilton-Poisson systems theory on jet bundle. Key–Words: fractional Stieltjes action, Euler-Lagrange or Hamilton equations, dynamic connection, symplectic manifold. 1 Euler-Lagrange ODEs associated to single-time Stieltjes actions Two functions f: R → R and g: R × R+ →