By Girish S. Setlur
The Countable and the UncountableReview of Lagrangian MechanicsThe Hamiltonian FormulationFlows and SymmetriesDynamics of a continual SystemVariational MethodsSymmetries and Noether's TheoremNoether's Theorem in a Lagrangian SettingNoether's Theorem in a Hamiltonian SettingDynamical SymmetriesSymmetries in box TheoriesThe Electromagnetic box and rigidity strength TensorRelativistic Nature of the ElectromagneticRead more...
summary: The Countable and the UncountableReview of Lagrangian MechanicsThe Hamiltonian FormulationFlows and SymmetriesDynamics of a continuing SystemVariational MethodsSymmetries and Noether's TheoremNoether's Theorem in a Lagrangian SettingNoether's Theorem in a Hamiltonian SettingDynamical SymmetriesSymmetries in box TheoriesThe Electromagnetic box and pressure strength TensorRelativistic Nature of the Electromagnetic FieldLagrangian of the EM FieldStress strength Tensor of the EM FieldSolution of Maxwell's Equations utilizing Green's FunctionsDiffraction TheoryElasticity concept and Fluid MechanicsStress
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Extra info for Dynamics of Classical and Quantum Fields : An Introduction
Example text
39) Thus the conjugate does not flow relative to the flow parameter. 40) ✐ ✐ ✐ ✐ ✐ ✐ “K15412” — 2013/10/31 — 10:54 ✐ ✐ The Countable and the Uncountable 13 when Q ≡ G, G is seen to be independent of the flow parameter α if G depends on α only through the phase space variables. We may see that Eq. 37) and Eq. 38) are consistent with the statement, G(q, p) = px y − py x. 41) This is nothing but the component of the angular momentum along the −z axis. Thus, angular momentum is conjugate to (also known as a generator of) simple rotations in coordinate space.
Try out all other possibilities. Successively transform N (you defined as T1 = ∂E should get a chemical potential somewhere) and V (you should get pressure). Try also transforming two variables together and finally all three of them. Can you recognize any of the transformed functions as the standard ones from thermodynamics textbooks? 4 Verify that the Lagrange equation of Eq. 94) is nothing but the familiar timedependent Schrodinger equation. Show that in this case, N(t) = dx ψ∗ (x,t)ψ(x,t) is independent of time.
94) is nothing but the familiar timedependent Schrodinger equation. Show that in this case, N(t) = dx ψ∗ (x,t)ψ(x,t) is independent of time. If we regard ψ(x,t) as a field obeying the classical field equations of this Lagrangian, then the statement that N(t) is time independent is true only in an average sense following Ehrenfest’s principle. 5 The Brachistochrone problem showed that the path which ensures the time of flight (Eq. 102)) an extremum is a cycloid. Prove that this extremum is a minimum (the second derivative should be positive).