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[NOTES/CM-11003] Action Angle Variables

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

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

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

[NOTES/CM-11004] Hamilton Jacobi Equation

 We derive the time dependent and time independent Hamilton Jacobi equations, amilton's characetrirstic function is introduced  as  solution of the time independent equation.

 kapoor's picture 24-05-25 21:05:33

[NOTES/CM-10003] Four Types of Canonical Transformations


A canonical transformation is a change of variables \((q,p) \rightarrow (Q,P)\) in phase space such that the Hamiltonian form of equations of motion is preserved. Depending choice of independent variables we have four special cases of canonical transformations., Generating functions for the four cases are introduced and details of the four cases are discussed.

 kapoor's picture 24-05-20 16:05:33

[NOTES/CM-10002] A Summary Finite Canonical Transformations

Important relations of four types of transformations are summarized.

 kapoor's picture 24-05-20 07:05:43

[NOTES/CM-10008] Examples --- Canonical Transformations

Several examples on canonical transformations are given.

 kapoor's picture 24-05-19 08:05:33

[NOTES/CM-10007] Generator of a Canonical Transformation

The definition  finite and infinitesimal canonical transformation are given. Using the action principle we define the generator of a canonical transformation.

 kapoor's picture 24-05-19 05:05:47

[NOTES/CM-10006] Two Simple Examples of Canonical Transformations

$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}$

 kapoor's picture 24-05-18 20:05:43

[NOTES/CM-09014] Angular Momentum of a Rigid Body

The angular momentum of a rigid  body is given by  where \(\mathbf I\) is moment of inertia tensor and \(\vec \omega \) is the angular velocity.\begin{eqnarray} \vec{L}&=&\int dv \rho(\vec{X})\vec{X}\times(\vec{\omega}\times\vec{X})\\ &=&\int dV \rho(\vec{X})\Big[(\vec{X}\cdot\vec{X})\vec{\omega}-(\vec{X}\cdot\vec{\omega} )\vec{X}\Big] \end{eqnarray} or \(\vec L=\mathbf I\, \vec \omega\).

 kapoor's picture 24-05-15 19:05:17

[NOTES/CM-09008] Specifying Orientation Using Body Axes

A possible way of specifying the orientation of a rigid body is to give orientation of body fixed axes w.r.t. a space fixed axes.  Euler angles are a  useful set generalized coordinates to specify orientation of the body axes relative to a space fixed axis.

 kapoor's picture 24-05-15 07:05:22

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