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Examples of application of Noether's theorem are given for mechanical systems. The following relationship between symmetry and corresponding conservation law is demonstrated  by means of explicit examples of system consisting of finite number of particles.

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Symmetry transformation is defined; statement and the proof of Noether's theorem is given for mechanics of several point particles.

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Some limitations of Newtonian mechanics are pointed out.

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Whenever the generalized force \(Q_k\) \begin{equation}\label{AB01} Q_k=\sum_\alpha F^\text{(e)}_\alpha \pp[\vec{x}_\alpha]{q_k}. \end{equation} can be written in the form \begin{equation}\label{AB02} Q_k = \dd{t}\Big(\pp[U]{\dot{q}_k}\Big)-\pp[U]{q_k} . \end{equation} the equations of motion take the Euler Lagrange form with \(L=T-U\). It is demonstrated that the Lorentz force,the force on a charged particle in e.m. fields, has the form as in \eqref{AB02}. An expression for the generalized potential \(U\) is derived.

If the Lagrangian does not depend on time explicitly, the Hamiltonian \(H=\sum_{k=1}\frac {\partial{L}}{\partial{\dot q_k}}\dot q_k-L\) is a constant of motion. For conservative systems of many particles, the Hamiltonian coincides with the total energy. is conserved.