Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 621of H preserves the constraint N and, restricted to N, it obeys the Hamiltonianequationγ⌋Ω N + i ∗ N dH = 0 (4.156)of a Dirac constraint system on N with a Hamiltonian H.The map (4.154) sends the vector–field γ (4.155) onto the vector–fieldγ T = ˙q α ∂ α + (∂ µ H∂ α ∂ µ H − ∂ µ H∂ α ∂ µ H) ˙∂ αon T Q. This vector–field defines the second–order dynamical equation¨q α = ∂ µ H∂ α ∂ µ H − ∂ µ H∂ α ∂ µ H (4.157)on Q which preserves the subbundle of hyperboloids (4.151).The following is a basic example of relativistic Hamiltonian systems.PutH = 12m gµν (p µ − b µ )(p ν − b ν ),where m is a constant and b µ dq µ is a covector–field on Q. Then H T =2m −1 H − 1 and {H, H T } = 0. The constraint H T = 0 defines a closedimbedded one-codimensional subbundle N of T ∗ Q. The Hamiltonian equation(4.156) takes the form γ⌋Ω N = 0. Its solution (4.155) reads˙q α = 1 m gαν (p ν − b ν ),ṗ α = − 12m ∂ αg µν (p µ − b µ )(p ν − b ν ) + 1 m gµν (p µ − b µ )∂ α b ν .The corresponding second–order dynamical equation (4.157) on Q is¨q α = Γ α µν ˙q µ ˙q ν − 1 m gλν F µν ˙q µ , (4.158)Γ α µν = − 1 2 gλβ (∂ µ g βν + ∂ ν g βµ − ∂ β g µν ), F µν = ∂ µ b ν − ∂ ν b µ .It is a geodesic equation with respect to the affine connectionK α µ = Γ α µν ˙q ν − 1 m gλν F µνof type (4.153). For example, let g be a metric gravitational field and letb µ = eA µ , where A µ is an electromagnetic potential whose gauge holdsfixed. Then the equation (4.158) is the well–known equation of motion ofa relativistic massive charge in the presence of these fields.