Ivancevic_Applied-Diff-Geom

928 Applied Differential Geometry: A Modern Introduction5.12 Application: Modern GeometrodynamicsIn this subsection we present some modern developments of the classicalEinstein–Wheeler geometrodynamics that we briefly reviewed as a motivationto our geometrical machinery.5.12.1 Stress–Energy–Momentum TensorsWhile in analytical mechanics there exists the conventional differential energyconservation law, in field theory it does not exist (see [Sardanashvily(1998)]). Let F be a smooth manifold. In time–dependent mechanics onthe phase–space R × T ∗ F coordinated by (t, y i , ẏ i ) and on the configurationspace R×T F coordinated by (t, y i , ẏ i ), the Lagrangian energy and theconstruction of the Hamiltonian formalism require the prior choice of a connectionon the bundle R × F −→ R. However, such a connection is usuallyhidden by using the natural trivial connection on this bundle. Therefore,given a Hamiltonian function H on the phase–space manifold R × T ∗ F , wehave the usual energy conservation lawdHdt ≈ ∂H∂t(5.384)where by ‘≈’ is meant the weak identity modulo the Hamiltonian equations.Given a Lagrangian function L on the configuration manifold R×T F , thereexists the fundamental identity∂L∂t + d dt (ẏi (t) ∂L − L) ≈ 0 (5.385)∂ẏi modulo the equations of motion. It is the energy conservation law in thefollowing sense. Let ̂L be the Legendre morphism given by ẏ i ◦ ̂L =∂ẏiL, and Q = Im ̂L the Lagrangian constraint manifold. Let H be aHamiltonian function associated with L and Ĥ the momentum morphism,ẏ i ◦ Ĥ = ∂ ẏ iH. Every solution r of the Hamiltonian equations of H whichlives on Q yields the solution Ĥ ◦ r of the Euler–Lagrangian equations ofL. Then, the identity (5.385) on Ĥ ◦ r recovers the energy transformationlaw (5.384) on r. 44 There are different Hamiltonian functions associated with the same singular Lagrangianfunction as a rule. Given such a Hamiltonian function, the Lagrangian constraintspace Q plays the role of the primary constraint space, and the Dirac procedurecan be used in order to get the final constraint space where a solution of the Hamiltonianequations exists [Gotay et. al. (1978); León and Marrero (1993)].