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Infinitesimal variation of the action functional is defined and computed for a an arbitrary path \(C\). It is shown that the requirement that the variation, with fixed end points, be zero is equivalent to the path \(C\) being the classical path in the configuration space.
Symmetries play an important role in many areas of Physics, Chemistry and Particle Physics.
Expression for the Lagrangian for a charged particle in electromagnetic field is given and the Euler Lagrange equations are shown to coincide with EOM with Lorentz force on the charged particle.
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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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The invariance of the action under time translations leads to conservation of Hamiltonian. This means that the Lagrangian should be independent of time for the law of energy conservation to hold.
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Gellman-Levi method for computing Noether charge associating with a symmetry transformation is explained, In case of a broken symmetry the Noether generator varies with time and its rate of variation can be computed in a simple manner by the and computing its time variation by this method.
Action principle is stated; Euler Lagrange EOM are obtained from the action principle.$\newcommand{\qbf}{\mathbf{q}}$
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