Generalized Buchdahl equations as Lie-Hamilton systems from the 'book' and oscillator algebras: Quantum deformations and their general solution

Abstract

We revisit the nonlinear second-order differential equations $$\ddot{x}(t)=a(x)\dot{x}(t{)}^2+b(t)\dot{x}(t)$$ where $a(x)$ and $b(t)$ are arbitrary functions on their argument from the perspective of Lie-Hamilton systems. For the particular choice $a(x)=3/x$ and $b(t)=1/t$, these equations reduce to the Buchdahl equation considered in the context of General Relativity. It is shown that these equations are associated to the 'book' Lie algebra ${\mathfrak{b}}_2$, determining a Lie-Hamilton system for which the corresponding $t$-dependent Hamiltonian and the general solution of the equations are given. The procedure is illustrated considering several particular cases. We also make use of the quantum deformation of ${\mathfrak{b}}_2$ with quantum deformation parameter $z$ (where $q={\mathrm{e}}^z$), leading to a deformed generalized Buchdahl equation. Applying the formalism of Poisson-Hopf deformations of Lie-Hamilton systems, we derive the corresponding deformed $t$-dependent Hamiltonian, as well as its general solution. The generalized Buchdahl equation is further extended to the oscillator Lie-Hamilton algebra ${\mathfrak{h}}_4\supset {\mathfrak{b}}_2$, together with its quantum deformation, and the corresponding (deformed) equations are also analyzed for their exact solutions. The presence of the quantum deformation parameter $z$ is interpreted as the introduction of an integrable perturbation of the (initial) generalized Buchdahl equation, which is described in detail in its linear approximation. Finally, it is also shown that, under quantum deformations, the higher-dimensional deformed generalized Buchdahl equations from either the book or the oscillator algebras do not reduce to a sum of copies of the initial system but to intrinsic coupled systems governed by $z$.

arXiv comment33 pages. New material covering extensions of the generalized Buchdahl equations from the (deformed) oscillator algebra has been included in sections 6, 7 and 8

References in corpus (15)

• On the complete integrability and linearization of certain second order nonlinear ordinary differential equations
• A direct approach to the construction of standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients
• Jacobi's Last Multiplier and Lagrangians for Multidimensional Systems
• Lie-Hamilton systems on the plane: properties, classification and applications
• Fermion-Boson Interactions and Quantum Algebras
• Non-standard quantum (1+1) Poincaré group: a $T$--matrix approach
• Lie bialgebra quantizations of the oscillator algebra and their universal $R$--matrices
• On the spectrum of a Hamiltonian defined on su_q(2) and quantum optical models
• Lie-Hamilton systems on the plane: Applications and superposition rules
• Boson representations, non-standard quantum algebras and contractions
• Poisson-Hopf algebra deformations of Lie-Hamilton systems
• Exact solutions and superposition rules for Hamiltonian systems generalizing time-dependent SIS epidemic models with stochastic fluctuations
• Poisson-Hopf deformations of Lie-Hamilton systems revisited: deformed superposition rules and applications to the oscillator algebra
• Generalized time-dependent SIS Hamiltonian models: Exact solutions and quantum deformations
• Solutions by quadratures of complex Bernoulli differential equations and their quantum deformation