In Chapter 2, we introduced the Lagrangian and the Hamiltonian equations of motion. The variational formulation of Chapter 7 describes the Lagrangian as an energy density functional from which it is possible to derive the equations of motion. In the case of wave propagation, the physics of nonlinear wave interactions becomes mathematically tractable when we use the Hamiltonian formalism with the understanding that the classical spin waves can be represented by their complex amplitudes instead of Bose operators that would represent magnons. The Hamiltonian method is specifically suitable for the analysis of weakly interacting and weakly dissipative wave systems, where nonlinear interactions can be treated as higher order corrections to the lowest order wave solutions. The Hamiltonian yields first-order differential equations which are easier to solve than Lagrange’s equations.
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Stancil, D.D., Prabhakar, A. (2009). Nonlinear Interactions. In: Spin Waves. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-77865-5_9
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