The study of conserved quantities and symmetries for two-dimensional isotropic harmonic charged oscillator moving in homogeneous magnetic field

The kinematic differentiation equations of two-dimensional isotropic harmonic charged oscillator moving in a homogeneous magnetic are obtained by using Newton’s second law. Two integrals (conserved quantities) are obtained by directly integrating the kinematic differentiation equations. The relationship between the Lagrangian and the conserved quantity is established through the Legendre transformation, thereby obtaining a Lagrangian function of the system. The Noether symmetry and Lie symmetry of the infinitesimal transformations corresponding to the conserved quantities are studied. Finally, the kinematical equations of the system are obtained.