I am investigating the problem of taking a hamiltonian of bilinear terms, and converting them into a bunch of uncoupled oscillators, such as in a periodic lattice. To do this, you have to introduce complex coordinates. My question is: why does the below still hold true?
Normally the Lagrange equations are derived using "real partial" derivatives, but of course the partial derivative with respect to a complex variable (like below) is defined as: 1/2( d/dx - i d/dy). I only have a very basic familiarity with the Hamiltonian-Lagrange formalism, but I guess my question is, is it true that the Hamiltonian equations can be extended to complex conjugate coordinates as seen below? If so, why? This seems to be a topic that is essentially nonexistent in the literature, and everywhere I go to investigate this topic of decoupled oscillators takes it as a given.