Inverse Lie Symmetry Analysis for Nonlinear Hamiltonian Systems on Time Scales

We investigate Lie symmetries and their associated inverse problems in the framework of Hamiltonian field theories formulated on time scales. Time scale calculus provides a unified setting for continuous and discrete dynamics, offering a natural description of nonlinear systems that exhibit hybrid temporal behavior. Within this context, we establish Hamilton’s canonical equations on time scales and derive the corresponding Lie symmetry transformations together with their infinitesimal generators. Based on invariance principles, we obtain conserved quantities and formulate necessary and sufficient conditions under which weak and strong Lie symmetries exist. We then address the inverse symmetry problem, namely how a prescribed invariant determines the underlying symmetry structure of nonlinear time-scale Hamiltonian systems. A nontrivial example is presented to illustrate the effectiveness of the developed theory. The results generalize classical Lie symmetry methods to the time-scale setting and contribute a new perspective to the analysis of nonlinear dynamics in regimes where discrete and continuous effects coexist.