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I Lesson Overview
Learning Goals
- Application of energy conservation law to solve problems in one dimension.
- Reducing the solution to quadratures.
- Define turning points and get classically accessible regions.
- Inferring general properties of motion, for different values of energy, from the graph of the potential.
Prerequisites
Familiarity with energy conservation; Integration of simple first order differential equations. Plotting graph of a given function. Using graphs.
Recall and Discuss
For conservative forces, the potential energy can be defined and one has the conservation law for total energy \(E\). This means as the body moves and passes though different positions, the energy does not change. Equivalently, \begin{equation} \dd{t} {\text{K.E. + P.E.}} =0. \end{equation} if we make use of equations.
II Main Lesson
\(\S\) 1 Reducing Solution to a Quadrature
\(\S\) 2 General Properties of Motion in One Dimension
\(\S\) 3 An Example of Using Plot of $ V(x) $
III EndNotes
- More examples in one dimension, problems and exercises.
- We have seen that the plot of potential energy can give a lot of useful information about the possible motion of the particle. The same 'story' can be repeated for many other cases using conservation laws. One such important case is system of two bodies interacting with a potential depending only on the distance between them. You may want to continue to Kepler problem of planetary motion.
- Lagrangian formalism offers ways of identifying conservation laws. Nother's theorem relates symmetries of the Lagrangian to conservation laws. A cyclic coordinates can be easily identified and leads to conservation of canonical momentum. Learn about cyclic cordinates; Noether's theorem about symmetries and conservation laws.
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