  This paper investigates the dynamics and integrability of the double spring
pendulum, which has great importance in studying nonlinear dynamics, chaos, and
bifurcations. Being a Hamiltonian system with three degrees of freedom, its
analysis presents a significant challenge. To gain insight into the system's
dynamics, we employ various numerical methods, including Lyapunov exponents
spectra, phase-parametric diagrams, and Poincar\'e cross-sections. The novelty
of our work lies in the integration of these three numerical methods into one
powerful tool. We provide a comprehensive understanding of the system's
dynamics by identifying parameter values or initial conditions that lead to
hyper-chaotic, chaotic, quasi-periodic, and periodic motion, which is a novel
contribution in the context of Hamiltonian systems. In the absence of
gravitational potential, the system exhibits $S^1$ symmetry, and the presence
of an additional first integral was identified using Lyapunov exponents
diagrams. We demonstrate the effective utilisation of Lyapunov exponents as a
potential indicator of first integrals and integrable dynamics. The numerical
analysis is complemented by an analytical proof regarding the non-integrability
of the system. This proof relies on the analysis of properties of the
differential Galois group of variational equations along specific solutions of
the system. To facilitate this analysis, we utilised a newly developed
extension of the Kovacic algorithm specifically designed for fourth-order
differential equations. Overall, our study sheds light on the intricate
dynamics and integrability of the double spring pendulum, offering new insights
and methodologies for further research in this field.
