In classical mechanics, both Newtonian mechanics and Lagrangian/Hamiltonian mechanics can describe the conservation of energy and momentum.
In Newtonian mechanics, the conservation of energy and momentum can be derived directly from Newton's laws of motion and the work-energy theorem. Newton's laws state that the net force acting on an object is equal to its mass times its acceleration (F = ma), and the work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. By considering systems where no external forces are present, and applying the principle of conservation of energy and momentum, one can deduce that energy and momentum are conserved in those systems.
Lagrangian and Hamiltonian mechanics provide an alternative formulation of classical mechanics based on the principle of least action. These formulations are based on the concept of a Lagrangian or a Hamiltonian function, respectively, which describe the dynamics of a system. Both Lagrangian and Hamiltonian mechanics are powerful frameworks for describing the conservation principles of energy and momentum.
In Lagrangian mechanics, the principle of least action leads to the Euler-Lagrange equations, which govern the motion of a system. The Lagrangian function incorporates the kinetic and potential energies of the system, and by applying Noether's theorem, one can show that the conservation of energy and momentum arises naturally from the symmetry properties of the Lagrangian.
Similarly, in Hamiltonian mechanics, the conservation of energy and momentum can be derived from the Hamiltonian function, which is closely related to the total energy of the system. The Hamiltonian equations of motion, derived from the Hamiltonian function, preserve energy and momentum throughout the system's evolution.
Therefore, principles of energy and principles of mechanics are intimately related in classical mechanics. The conservation of energy and momentum can be derived from both Newtonian mechanics and Lagrangian/Hamiltonian mechanics, providing a consistent and interconnected framework for understanding the behavior of physical systems.