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Date  |
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[LECS/CM-03] Action Principles |
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24-06-11 14:06:35 |
[NOTES/CM-04006] Hamiltonian for a Charged Particle$\newcommand{\Label}[1]{\label{#1}}\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}\newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}$ The Hamiltonian of a charged particle in electromagnetic field is derived starting from the Lagrangian. |
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24-06-09 18:06:19 |
[NOTES/CM-04005] Poisson Bracket Properties$\newcommand{\Label}[1]{\label{#1}} We list important properties of Poisson brackets. Poisson bracket theoerem statement that the Poisson bracket of two integrals of motion is again an integral of motion, and its proof is given. A generalization of the theorem is given without proof. |
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24-06-09 18:06:28 |
[NOTES/CM-04003] Variational Principles in Phase Space$\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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24-06-09 18:06:51 |
[NOTES/CM-04001] Hamiltonian Formulation of Classical Mechanics$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}\newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}} Transition from the Lagrangian to Hamiltonian formalism is described; Hamiltonian equations of motion are obtained. |
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24-06-09 04:06:25 |
[NOTES/CM-03008] Hamilton's Principle$\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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24-06-06 12:06:18 |
[NOTES/CM-03007] Symmetries --- Numerous Applications to Different Areas.Symmetries play an important role in many areas of Physics, Chemistry and Particle Physics. |
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24-06-05 12:06:53 |
[NOTES/CM-02008] Eliminating Cyclic Coordnatescyclic 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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24-06-03 08:06:27 |
[NOTES/CM-11001] Hamilton's Principal Function$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}$ 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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24-06-02 23:06:17 |
[NOTES/CM-11005] Periodic motionFor 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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24-05-31 16:05:11 |