Constrained Lagrangians and Hamiltonians, Classical and Quantum Neural Networks, Examples of Inaccessibility, Hawking Radiation, Conditioning in Quantum Mechanics
Inge Svein Helland, Harish Parthasarathy · 2025
Consider for example the pde https://www.w3.org/1998/Math/MathML" display="block"> ∂ t ϕ ( t , r ) = L ( ϕ ( t , r ) ) where L is a linear partial differential operator expressed as https://www.w3.org/1998/Math/MathML" display="block"> L ( ϕ ( t , r ) ) = ∑ k ≥ 1 A k ( t , r ) ( ∇ ⊗ k ⊗ ϕ ( t , r ) ) with ϕ being a vector valued field and A k ( t, r ) matrices of appropriate size. These dynamical equations can be derived from the variational principle https://www.w3.org/1998/Math/MathML" display="block"> δ S [ ϕ , Λ ] = 0 where https://www.w3.org/1998/Math/MathML" display="block"> S [ ϕ , Λ ] = ∫ Λ ( x ) T ( ∂ t ϕ ( x ) − L ( ϕ ( x ) ) ) d 4 x = ∫ ℒ d x