This paper develops a way to write Lagrangians and Hamiltonians for a class of “spatially nonlocal” field theories. These are models where the equations at a point x depend not only on the field at x but also on the field at the reflected point −x. The authors present a generalized variational and Hamiltonian framework that keeps the familiar algebraic structure of classical field theory while accounting for reflected-field contributions.

At the core of the work is a modified variational principle. When the Lagrangian density depends on both the field at x and the field at −x, the usual Euler–Lagrange equations pick up extra terms from the reflected field. The paper derives these generalized Euler–Lagrange equations and introduces a generalized functional derivative that adds a reflected-field term to the usual variational derivative. That change is then used to extend canonical constructions, such as Poisson brackets and Hamilton’s equations, to the nonlocal setting by replacing ordinary functional derivatives with the generalized ones.

The authors apply the formalism to three concrete nonlocal nonlinear Schrödinger (NLS) equations. For the Ablowitz–Musslimani (AM) equation, which has parity symmetry and takes the form i q_t − q_xx + 2 q^2 q^*(−x) = 0, they construct what they state is the first standard Lagrangian density written directly in terms of the complex field and its reflected complex conjugate. Although that Lagrangian looks complex, the authors show it becomes real when the field is split into real and imaginary parts. They then use the Dirac–Bergmann algorithm for constrained systems to build a complete Hamiltonian formulation for the AM equation.

The same generalized Hamiltonian framework is used for two nonlocal NLS systems introduced by Velasco‑Juan and Fujioka, called LN1 and LN2. For those cases the method reproduces the corresponding generalized Euler–Lagrange equations and gives consistent canonical variables, primary constraints, and Hamiltonians. Taken together, these examples show the formalism can treat different classes of spatially nonlocal NLS equations in a unified way.