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$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}$
1. Hamilton Jacobi Theory
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1.1 Hamilton's Principal Function 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. |
| 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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We derive the time dependent and time independent Hamilton Jacobi equations, amilton's characetrirstic function is introduced as solution of the time independent equation. |
2 Frequencies of Bounded Motion
| 2.1 Periodic motion
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. |
| 2.2 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. |
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4727:DIamond Point
