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[LECS/CM-11] Hamilton Jacobi Theory

Node id: 6291page
kapoor's picture 24-06-11 16:06:32 n

[LECS/CM-03] Action Principles

Node id: 6275page
kapoor's picture 24-06-11 14:06:35 n

[LECS/CM-04003]

Node id: 6285page
kapoor's picture 24-06-10 04:06:56 n

[LECS/CM-04002]

Node id: 6284page
kapoor's picture 24-06-10 04:06:48 n

[NOTES/CM-04006] Hamiltonian for a Charged Particle

Node id: 6280page

The Hamiltonian of a charged particle in electromagnetic field is derived starting from the Lagrangian.

kapoor's picture 24-06-09 18:06:19 n

[NOTES/CM-04005] Poisson Bracket Properties

Node id: 6279page

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.

kapoor's picture 24-06-09 18:06:28 n

[NOTES/CM-04003] Variational Principles in Phase Space

Node id: 6278page

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.

kapoor's picture 24-06-09 18:06:51 n

[NOTES/CM-04002] Poisson Bracket Formalism

Node id: 6277page

We 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.

kapoor's picture 24-06-09 16:06:26 n

[NOTES/CM-04007] Gauge Invariance of Hamiltonian for a Charged Particle

Node id: 6281page

The 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 \[\vec p \longrightarrow \vec p^\prime -(e/c) \nabla \Lambda.\]

kapoor's picture 24-06-09 10:06:00 n

[NOTES/CM-04004] Summary of Lagrangian and Hamiltonian Formalisms

Node id: 6131page

A  summary of Lagrangian and Hamiltonian formalisms is given in tabular form.

kapoor's picture 24-06-09 06:06:48 n

[NOTES/CM-04001] Hamiltonian Formulation of Classical Mechanics

Node id: 6276page

Transition from the Lagrangian to Hamiltonian formalism is described; Hamiltonian equations of motion are obtained.

kapoor's picture 24-06-09 04:06:25 n

[NOTES/CM-03008] Hamilton's Principle

Node id: 6126page

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.

kapoor's picture 24-06-06 12:06:18 n

[NOTES/CM-03007] Symmetries --- Numerous Applications to Different Areas.

Node id: 6125page

Symmetries play an important role in many areas of Physics, Chemistry and Particle Physics.

kapoor's picture 24-06-05 12:06:53 n

[NOTES/CM-02008] Eliminating Cyclic Coordnates

Node 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.

kapoor's picture 24-06-03 08:06:27 n

[NOTES/CM-02002] Constrraints, Degrees of Freedom, Generalized Coordinates

Node id: 6101page

The 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.

kapoor's picture 24-06-03 05:06:19 n

[NOTES/CM-11001] Hamilton's Principal Function

Node id: 6264page

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.

kapoor's picture 24-06-02 23:06:17 n

[NOTES/CM-11005] Periodic motion

Node id: 6271page

For 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.

kapoor's picture 24-05-31 16:05:11 n

[NOTES/CM-11003] Action Angle Variables

Node id: 6266page

The action angle variables are defined in terms of solution of Hamilton-Jacobi 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.

kapoor's picture 24-05-31 15:05:46 n

[NOTES/CM-11002] Jacobi's Complete Integral

Node id: 6265page

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}

kapoor's picture 24-05-29 06:05:31 n

[LECS/QFT-ALL] Quantum Field Theory --- Collection of No-Frills Lectures

Node id: 6211collection

About 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.

kapoor's picture 24-05-29 04:05:04 n

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