
[LECS/CM11] Hamilton Jacobi TheoryNode id: 6291page$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}$ 

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[LECS/CM03] Action PrinciplesNode id: 6275page 

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[LECS/CM04003] Node id: 6285page 

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[LECS/CM04002] Node id: 6284page 

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[NOTES/CM04006] Hamiltonian for a Charged ParticleNode id: 6280page$\newcommand{\Label}[1]{\label{#1}}\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}\newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}$
The Hamiltonian of a charged particle in electromagnetic field is derived starting from the Lagrangian. 

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[NOTES/CM04005] Poisson Bracket PropertiesNode id: 6279page$\newcommand{\Label}[1]{\label{#1}} \newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}\newcommand{\mHighLight}[1]{\mbox{\boxed{\text{#1}}}}$
We list important properties of Poisson brackets. Poisson bracket theoerem statement that the Poisson bracket of two integrals of motion is again an integral of motion, and its proof is given. A generalization of the theorem is given without proof. 

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[NOTES/CM04003] Variational Principles in Phase SpaceNode id: 6278page$\newcommand{\Prime}{{^\prime}} \newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}\newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}\newcommand[\Pime][{^\prime}]\newcommand{\qbf}{{\mathbf q}}\newcommand{\pbf}{\mathbf p}$
In the canonical formulation of mechanics, the state of a system is represented by a point in phase space. As time evolves, the system moves along a path in phase space. Principle of least action is formulated in phase space. The Hamilton's equations motion follow if we demand that, for infinitesimal variations, with coordinates at the end point fixed, the action be extremum. No restrictions on variations in momentum are imposed. 

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[NOTES/CM04002] Poisson Bracket FormalismNode id: 6277pageWe define Poisson bracket and the Hamiltonian equations of motion are written in terms of Poisson brackets. The equation for time evolution of a dynamical variable is written in terms of Poisson brackets. It is proved that a dynamical variable \(F\), not having explicit time dependence, is constant of motion if its Poisson bracket with the Hamiltonian is zero.
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[NOTES/CM04007] Gauge Invariance of Hamiltonian for a Charged ParticleNode id: 6281pageThe Hamiltonian for a charged particle in electromagnetic field is invariant under the gauge transformation \[\vec A \longrightarrow \vec A^\prime = \vec A \nabla \Lambda\]if the canonical momentum is taken to transform as \[\vec p \longrightarrow \vec p^\prime (e/c) \nabla \Lambda.\] $\newcommand{\Prime}{{^\prime}}\newcommand{\Label}[1]{\label{#1}}$ 

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[NOTES/CM04004] Summary of Lagrangian and Hamiltonian FormalismsNode id: 6131pageA summary of Lagrangian and Hamiltonian formalisms is given in tabular form. 

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[NOTES/CM04001] Hamiltonian Formulation of Classical MechanicsNode id: 6276page$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}\newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}} \newcommand{\Label}[1]{\label{#1}}\newcommand{\qbf}{{\mathbf q}}\newcommand{\pbf}{{\mathbf p}}$
Transition from the Lagrangian to Hamiltonian formalism is described; Hamiltonian equations of motion are obtained. 

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[NOTES/CM03008] Hamilton's PrincipleNode id: 6126page$\newcommand{\Prime}{^\prime}\newcommand{\qbf}{{\bf q}}\newcommand{\lefteqn}{}$
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. 

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[NOTES/CM03007] Symmetries  Numerous Applications to Different Areas.Node id: 6125pageSymmetries play an important role in many areas of Physics, Chemistry and Particle Physics. 

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[NOTES/CM02008] Eliminating Cyclic CoordnatesNode id: 6048page cyclic coordinates and conjugate momentum can be completely eliminated following a procedure given by Ruth. The resulting dynamics is again formulated in terms of the remaining coordinates. 

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[NOTES/CM02002] Constrraints, Degrees of Freedom, Generalized CoordinatesNode id: 6101pageThe equations of motion in Newtonian mechanics are always written in Cartesian coordinates. If there are constraints among the coordinates, these have to be taken into account separately. A special type of constraints, called holonomic constraints and the number of degrees of freedom is defined. 

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[NOTES/CM11001] Hamilton's Principal FunctionNode id: 6264page$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}$
The Hamilton's principal function is defined as action integral \[S(q,t;q_0,t_0)=\int_{t_0}^t L dt\] expressed in terms of the coordinates and times, (q,t;q_0,t_0) at the end points. Knowledge of Hamilton's principal function is equivalent to knowledge of solution of the equations of motion. 

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[NOTES/CM11005] Periodic motionNode id: 6271pageFor a periodic system two types of motion are possible. In the first type both coordinates and momenta are periodic functions of time. An example of this type is motion of a simple pendulum. In the second type of motion only the coordinates are periodic functions of time. An example of the second type of motion is the conical pendulum where the angle keeps increasing, but the momentum is a periodic function of time. 

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[NOTES/CM11003] Action Angle VariablesNode id: 6266page$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$
The action angle variables are defined in terms of solution of HamiltonJacobi equation}. The application of action angle variables to computation of frequencies of bounded periodic motion is explained. An advantage offered by use of action angle variables is that the full solution of the equations of motion is not required. 

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[NOTES/CM11002] Jacobi's Complete IntegralNode id: 6265page$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}$
Jacobi's complete integral is defined as the action integral expressed in terms of non additive constants of motion and initial and final times. Knowledge of the complete integral is equivalent to the knowledge of the solution of equations of motion. Its relation with the Hamilton's principal function is \begin{equation} \pp[S_J (q, \alpha, t)]{\alpha_k} \pp[S_J (q_0, \alpha, t_0)]{\alpha_k} =0 \end{equation} 

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[LECS/QFTALL] Quantum Field Theory  Collection of NoFrills LecturesNode id: 6211collectionAbout this collection: This is a collection of Lecture Notes on Electromagnetic Theory. An effort is made to keep the content focused on the main topics. There is no discussion of related topics and no digression into unnecessary details.
Who may find it useful: Any one who wants to learn, or refresh all topics, in standard one semester Quantum Field Theory course.
Topics covered: The list of topics covered appears in the main body of this page. Click on any topic to see details and links to content pages. 

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