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Theoretical analysis reveals Noether symmetries and conserved quantities in non-standard Lagrangian systems, highlighting unified conservation laws across continuous and discrete dynamics.
In this paper, the fractional time-scales calculus is used to study the Noether symmetries of a non-standard Lagrangian system. First, we derive the corresponding equations of motion for exponential, logarithmic and power-law Lagrangians respectively based on the Caputo Δ derivatives. Next, we introduce the infinitesimal transformation, and discuss the Noether symmetry and conserved quantity of the non-standard Lagrangian system by using the invariance of the Hamilton action. Finally, we provide numerical examples to verify the conserved quantities.
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