Derivation of the Equations of Motion
In this document we explain how to derive the equations of motion from the Lagrangian
\[\begin{aligned} L &= \dot{x}_c\,\tilde{p}_q - \dot{p}_c\,\tilde{x}_q - \Biggl\{\delta + \frac{\chi}{2} \Bigl(x_c^2 + p_c^2 + \tilde{x}_q^2 + \tilde{p}_q^2 \Bigr) \Biggr\}(x_c\,\tilde{x}_q + p_c\,\tilde{p}_q) \\ &\quad + \kappa (x_c\,\tilde{p}_q - p_c\,\tilde{x}_q) + i \kappa (\tilde{x}_q^2 + \tilde{p}_q^2) + 2 \varepsilon\,\tilde{p}_q\,. \end{aligned}\]1. Recognizing the Canonical Structure
Notice that the Lagrangian is first order in time derivatives, with the kinetic (or “symplectic”) part given by
\[L_{\rm kin} = \dot{x}_c\,\tilde{p}_q - \dot{p}_c\,\tilde{x}_q\,.\]This form identifies the canonical pairs as:
- $x_c$ with conjugate momentum $\tilde{p}_q$, and
- $p_c$ with conjugate momentum $-\tilde{x}_q$.
Thus, the Lagrangian can be rewritten as
\[L = \tilde{p}_q\,\dot{x}_c - \tilde{x}_q\,\dot{p}_c - H(x_c, p_c, \tilde{x}_q, \tilde{p}_q)\,,\]where the “Hamiltonian” is given by
\[\begin{aligned} H(x_c, p_c, \tilde{x}_q, \tilde{p}_q) = &\; \Biggl[\delta + \frac{\chi}{2} \Bigl(x_c^2+p_c^2+\tilde{x}_q^2+\tilde{p}_q^2\Bigr)\Biggr](x_c\,\tilde{x}_q+p_c\,\tilde{p}_q) \\ &\; -\kappa(x_c\,\tilde{p}_q-p_c\,\tilde{x}_q) - i\kappa(\tilde{x}_q^2+\tilde{p}_q^2) - 2\varepsilon\,\tilde{p}_q\,. \end{aligned}\]2. Writing Hamilton’s Equations
Since the kinetic term is in canonical form, the Euler–Lagrange equations for our variables are equivalent to Hamilton’s equations. In particular, we have
\[\begin{aligned} \dot{x}_c &= \frac{\partial H}{\partial \tilde{p}_q}\,, &\quad \dot{\tilde{p}}_q &= -\frac{\partial H}{\partial x_c}\,,\\[1mm] \dot{p}_c &= -\frac{\partial H}{\partial \tilde{x}_q}\,, &\quad \dot{\tilde{x}}_q &= \frac{\partial H}{\partial p_c}\,. \end{aligned}\]3. Computing the Partial Derivatives
To compute these derivatives efficiently, one may introduce the shorthand
\[A \equiv \delta+\frac{\chi}{2}\Bigl(x_c^2+p_c^2+\tilde{x}_q^2+\tilde{p}_q^2\Bigr) \quad \text{and} \quad B \equiv x_c\,\tilde{x}_q+p_c\,\tilde{p}_q\,.\]Derivative with Respect to $\tilde{p}_q$:
\[\frac{\partial H}{\partial \tilde{p}_q} = \chi\,\tilde{p}_q\,B + A\,p_c - \kappa\,x_c - 2i\kappa\,\tilde{p}_q - 2\varepsilon\,.\]Thus, the equation of motion for $x_c$ is
\[\dot{x}_c = \chi\,\tilde{p}_q\,B + A\,p_c - \kappa\,x_c - 2i\kappa\,\tilde{p}_q - 2\varepsilon\,.\]Derivative with Respect to $x_c$:
\[\frac{\partial H}{\partial x_c} = \chi\,x_c\,B + A\,\tilde{x}_q - \kappa\,\tilde{p}_q\,.\]Thus, the equation of motion for $\tilde{p}_q$ is
\[\dot{\tilde{p}}_q = -\chi\,x_c\,B - A\,\tilde{x}_q + \kappa\,\tilde{p}_q\,.\]Derivative with Respect to $\tilde{x}_q$:
\[\frac{\partial H}{\partial \tilde{x}_q} = \chi\,\tilde{x}_q\,B + A\,x_c + \kappa\,p_c - 2i\kappa\,\tilde{x}_q\,.\]Thus, the equation of motion for $p_c$ is
\[\dot{p}_c = -\chi\,\tilde{x}_q\,B - A\,x_c - \kappa\,p_c + 2i\kappa\,\tilde{x}_q\,.\]Derivative with Respect to $p_c$:
\[\frac{\partial H}{\partial p_c} = \chi\,p_c\,B + A\,\tilde{p}_q + \kappa\,\tilde{x}_q\,.\]Thus, the equation of motion for $\tilde{x}_q$ is
\[\dot{\tilde{x}}_q = \chi\,p_c\,B + A\,\tilde{p}_q + \kappa\,\tilde{x}_q\,.\]4. Final Set of Equations
Summarizing, the equations of motion are
\[\boxed{ \begin{aligned} \dot{x}_c &= \chi\,\tilde{p}_q\,(x_c\,\tilde{x}_q+p_c\,\tilde{p}_q) +\left(\delta+\frac{\chi}{2}(x_c^2+p_c^2+\tilde{x}_q^2+\tilde{p}_q^2)\right)p_c -\kappa\,x_c-2i\kappa\,\tilde{p}_q-2\varepsilon\,,\\[1mm] \dot{\tilde{p}}_q &= -\chi\,x_c\,(x_c\,\tilde{x}_q+p_c\,\tilde{p}_q) -\left(\delta+\frac{\chi}{2}(x_c^2+p_c^2+\tilde{x}_q^2+\tilde{p}_q^2)\right)\tilde{x}_q +\kappa\,\tilde{p}_q\,,\\[1mm] \dot{p}_c &= -\chi\,\tilde{x}_q\,(x_c\,\tilde{x}_q+p_c\,\tilde{p}_q) -\left(\delta+\frac{\chi}{2}(x_c^2+p_c^2+\tilde{x}_q^2+\tilde{p}_q^2)\right)x_c -\kappa\,p_c+2i\kappa\,\tilde{x}_q\,,\\[1mm] \dot{\tilde{x}}_q &= \chi\,p_c\,(x_c\,\tilde{x}_q+p_c\,\tilde{p}_q) +\left(\delta+\frac{\chi}{2}(x_c^2+p_c^2+\tilde{x}_q^2+\tilde{p}_q^2)\right)\tilde{p}_q +\kappa\,\tilde{x}_q\,. \end{aligned} }\]