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1. Variational Principles in Mechanics
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1.1 Defining Action Functional Action principle is stated; Euler Lagrange EOM are obtained from the action principle. 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. Weiss action principle states that general iinfinitesimal variations in action functional about a path give an expression which depends on the end point if and only if the path is a classical path. |
2. Symmetries, Noether's Theorem Conservation Laws
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2.1 Symmetries and Conservation Laws Symmetry transformation is defined; statement and the proof of Noether's theorem is given for mechanics of several point particles. 2.2 A short cut to Noether generator and equation for its time variation 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. 2.3 Applications of Noether's Theorem 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. 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. |
3 Charged Particle In Electromagnetic Field
| 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. |
4.Symmetries --- Numerous Applications to Different Areas.
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An overview of role played by symmetries and conservation laws is areas of Physics, Chemistry and Particle Physics is given. |
