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We study the physical origins of parallel guiding center drift and parallel velocity-dependent effective magnetic field (B‖*) appearing in the Hamiltonian formulation of a particle motion in a curved magnetic field. The magnetic twist term is shown to be responsible for both of them. The parallel guiding center drift originates from the change in the effective parallel velocity due to the combined effect of the initial perpendicular motion (v⊥0) and the magnetic twist. The effective magnetic field arises from the collated effect of the change in v⊥0 due to its initial parallel guiding center motion and the conservation of magnetic moment. This understanding clearly shows that the appearance of B‖* is a consequence of magnetic moment conservation in curved magnetic fields.