Abstract We show that we can obtain expressions for conserved quantities directly from Newton’s second law. These conserved quantities coincide with the ones that are obtained with the aid of Noether’s theorem in the Lagrangian formalism. In particular, we consider the motion of a charged particle in an inhomogeneous electromagnetic field and we find constants of motion associated with the symmetries of the electromagnetic field, which are explicitly gauge-invariant. These constants of motion lead directly to conserved operators in the quantum formulation of the problem.

The Lie group method is a powerful technique for obtaining analytical solutions for various nonlinear differential equations. This study aimed to explore the behavior of nonlinear elastic wave equations and their underlying physical properties using Lie group invariants. We derived eight-dimensional symmetry algebra for the (3+1)-dimensional nonlinear elastic wave equation, which was used to obtain the optimal system. Group-invariant solutions were obtained using this optimal system. The same analysis was conducted for the damped version of this equation. For the conservation laws, we applied Noether's theorem to the nonlinear elastic wave equations owing to the availability of a classical Lagrangian. However, for the damped version, we cannot obtain a classical Lagrangian, which makes Noether's theorem inapplicable. Instead, we used an extended approach based on the concept of a partial Lagrangian to uncover conservation laws. Both techniques account for the conservation laws of linear momentum and energy within the model. These novel approaches add an application of variational calculus to the existing literature. This offers valuable insights and potential avenues for further exploration of the elastic wave equations.

A generalized Noether theorem is presented, relating symmetries and (equivalence classes of local) conservation laws in classical field theories; this is contrasted with the standard theorem. The concept of a “Noether” field theory is introduced, being a theory for which the generalized theorem applies; not only does this include the cases of Lagrangian and Hamiltonian field theories, these structures are “derived” from the Noether property in a natural way. The generalized theorem applies to currents and symmetries that contain derivatives of the fields up to an arbitrarily high order.