1. 9/14/2015PHY 711 Fall 2015 -- Lecture 91 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 9: Continue reading Chapter 3 & 6 1.Summary & review 2.Lagrange’s equations with constraints

2. 9/14/2015PHY 711 Fall 2015 -- Lecture 92

3. 9/14/2015PHY 711 Fall 2015 -- Lecture 93 Comment on problem Lagrangian formulation of Brachistochrone motion: s

4. 9/14/2015PHY 711 Fall 2015 -- Lecture 94

5. 9/14/2015PHY 711 Fall 2015 -- Lecture 95 Comments on generalized coordinates: Here we have assumed that the generalized coordinates q  are independent. Now consider the possibility that the coordinates are related through constraint equations of the form: Lagrange multipliers

6. 9/14/2015PHY 711 Fall 2015 -- Lecture 96 Simple example: u x y  

7. 9/14/2015PHY 711 Fall 2015 -- Lecture 97 Force of constraint; normal to incline

8. 9/14/2015PHY 711 Fall 2015 -- Lecture 98 Rational for Lagrange multipliers

9. 9/14/2015PHY 711 Fall 2015 -- Lecture 99 Euler-Lagrange equations with constraints: Example:  r mg

10. 9/14/2015PHY 711 Fall 2015 -- Lecture 910 Example continued:

11. 9/14/2015PHY 711 Fall 2015 -- Lecture 911 Another example:

12. 9/14/2015PHY 711 Fall 2015 -- Lecture 912 Another example: A particle of mass m starts at rest on top of a smooth fixed hemisphere of radius R. Find the angle at which the particle leaves the hemisphere. R 

13. 9/14/2015PHY 711 Fall 2015 -- Lecture 913 Example continued

14. 9/14/2015PHY 711 Fall 2015 -- Lecture 914 Example continued

15. 9/14/2015PHY 711 Fall 2015 -- Lecture 915 Consider a particle of mass m moving frictionlessly on a parabola z=c(x 2 +y 2 ) under the influence of gravity. Find the equations of motion, particularly showing stable circular motion.

16. 9/14/2015PHY 711 Fall 2015 -- Lecture 916

17. 9/14/2015PHY 711 Fall 2015 -- Lecture 917 Stable solution when these terms add to 0:

18. 9/14/2015PHY 711 Fall 2015 -- Lecture 918 Analysis of stable (circular) motion