The geometry of relativistic rheonomic Lagrange spaces (Q2770387)

The classical rheonomic Lagrange space \(L^n=(M,L(t,x,y))\) with time-dependent Lagrangian \(L:R\times TM\to R\) is considered in the book by \textit{R. Miron} and \textit{M. Anastasiei} [The Geometry of Lagrange Spaces, Fundamental Theories of Physics. 59, Kluwer (1994; Zbl 0831.53001)], where for an \(n\)-dimensional smooth manifold \(M\) the space \(R\times TM\) is regarded as a fibre bundle over \(M\). In the present paper the 1-jet vector bundle \(R\times TM\equiv J^1(R,M)\) is regarded as a fibre bundle over \(R\times M\). Consequently, the gauge group does not ignore the temporal reparametrizations.NEWLINENEWLINENEWLINEA pair \(RL^n=(J^1(R,M),L)\) with a Kronecker \(h\)-regular Lagrangian function \(L:J^1(T,M)\to R\) is called a relativistic rheonomic Lagrange space; here \(h=(h_{11})\) is the temporal semi-Riemannian metric and the vertical fundamental metrical \(d\)-tensor of \(L\) is defined by \({1\over 2}{\partial^2L \over\partial y^i\partial y^j}= h^{11}(t)g_{ij} (t,x,y)\). Its canonical nonlinear connection \(\Gamma\) as well as its canonical \(\Gamma\)-linear Cartan connection are constructed. The relativistic rheonomic Lagrangian electromagnetism with its Maxwell equations as well as the relativistic rheonomic Lagrangian gravitational theory are presented.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00064].