Ivancevic_Applied-Diff-Geom

618 Applied Differential Geometry: A Modern IntroductionLet N be coisotropic. Then a solution exists if the Poisson bracket {H, f}vanishes on N whenever f is a function vanishing on N. It is the Hamiltonianvector–field of H on Z restricted to N [Sardanashvily (2003)].Recall that a configuration space of non–relativistic time–dependentmechanics (henceforth NRM) of m degrees of freedom is an (m+1)D smoothfibre bundle Q −→ R over the time axis R [Mangiarotti and Sardanashvily(1998); Sardanashvily (1998)]. It is coordinated by (q α ) = (q 0 , q i ), whereq 0 = t is the standard Cartesian coordinate on R. Let T ∗ Q be the cotangentbundle of Q equipped with the induced coordinates (q α , p α = ˙q α ) withrespect to the holonomic coframes {dq α }. The cotangent bundle T ∗ Q playsthe role of a homogeneous momentum phase–space of NRM, admitting thecanonical symplectic formΩ = dp α ∧ dq α . (4.144)Its momentum phase–space is the vertical cotangent bundle V ∗ Q of theconfiguration bundle Q −→ R, coordinated by (q α , q i ). A Hamiltonian Hof NRM is defined as a section p 0 = −H of the fibre bundle T ∗ Q −→ V ∗ Q.Then the momentum phase–space of NRM can be identified with the imageN of H in T ∗ Q which is the one-codimensional (consequently, coisotropic)imbedded submanifold given by the constraintH T = p 0 + H(q α , p k ) = 0.Furthermore, a solution of a non–relativistic Hamiltonian system with aHamiltonian H is the restriction γ to N ∼ = V ∗ Q of the Hamiltonian vector–field of H T on T ∗ Q. It obeys the equation γ⌋Ω N = 0 [Mangiarotti andSardanashvily (1998); Sardanashvily (1998)]. Moreover, one can show thatgeometrical quantization of V ∗ Q is equivalent to geometrical quantizationof the cotangent bundle T ∗ Q where the quantum constraint ĤT ψ = 0on sections ψ of the quantum bundle serves as the Schrödinger equation[Sardanashvily (2003)].A configuration space of relativistic mechanics (henceforth RM) is anoriented pseudo–Riemannian manifold (Q, g), coordinated by (t, q i ). Itsmomentum phase–space is the cotangent bundle T ∗ Q provided with thesymplectic form Ω (4.144). Note that one also considers another symplecticform Ω + F where F is the strength of an electromagnetic field [Sniatycki(1980)]. A relativistic Hamiltonian is defined as a smooth real functionH on T ∗ Q [Mangiarotti and Sardanashvily (1998); Sardanashvily (1998)].Then a relativistic Hamiltonian system is described as a Dirac constraint