Bistable multilayered plates have attracted significant attention as morphing and adaptive structures, renowned for their unprecedented exceptional performance. However, accurately predicting both their shape and internal stresses poses formidable challenges due to their inherently geometrically nonlinear nature. For the first time, this paper introduces a Hamiltonian formalism aimed at achieving high-resolution analysis of nonlinear multilayered rectangular plates subjected to non-mechanical stimuli, including hygro-thermo-electro-magneto-elastic responses. A canonical system is strategically developed to compute membrane behaviors, yielding a set of dual differential equations for in-plane field variables. This method elegantly decouples out-of-plane variables from the full-state vector, leading to an analytical solution in the membrane problem via the technique of symplectic eigen-expansion. To predict the bending behaviors, the first variation of the Hamiltonian energy density function, expressed as a power series of the transverse deflection, ensures flexural equilibrium. The power series effectively captures all admissible out-of-plane deformations, enabling a smooth transition between linear and nonlinear plate responses, including bifurcation and limit points. The validity and accuracy of the proposed method are rigorously assessed through convergence tests and satisfaction of traction-free boundary conditions. The complete solution is evaluated across various configurations, ranging from monostability to bistability. It is cross-verified with high-fidelity finite-element methods (FEM), showing excellent agreements in both deformations and stress resultants. This research presents a physics-based methodology that unveils a parametric interplay between in-plane and out-of-plane field variables, curvatures, geometrically nonlinear strains, and inelastic external loadings. The framework serves as an efficient approach for comprehensively analyzing adaptive and morphing structures.