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PHY 7 11 Classical Mechanics and Mathematical Methods 9-9:50 AM MWF Olin 107

In this lecture, we will delve into Lagrange's equations with constraints, discussing the use of Lagrange multipliers and Euler-Lagrange equations. Examples involving particles on surfaces and in gravitational fields will be explored to illustrate these concepts.

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PHY 7 11 Classical Mechanics and Mathematical Methods 9-9:50 AM MWF Olin 107

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  1. PHY 711 Classical Mechanics and Mathematical Methods • 9-9:50 AM MWF Olin 107 • Plan for Lecture 9: • Continue reading Chapter 3 & 6 • Summary & review • Lagrange’s equations with constraints PHY 711 Fall 2017 -- Lecture 9

  2. PHY 711 Fall 2017 -- Lecture 9

  3. PHY 711 Fall 2017 -- Lecture 9

  4. PHY 711 Fall 2017 -- Lecture 9

  5. Comments on generalized coordinates: Here we have assumed that the generalized coordinates qs are independent. Now consider the possibility that the coordinates are related through constraint equations of the form: Lagrange multipliers PHY 711 Fall 2017 -- Lecture 9

  6. Simple example: u q y x q PHY 711 Fall 2017 -- Lecture 9

  7. Force of constraint; normal to incline PHY 711 Fall 2017 -- Lecture 9

  8. Rational for Lagrange multipliers PHY 711 Fall 2017 -- Lecture 9

  9. Euler-Lagrange equations with constraints: Example: r q mg PHY 711 Fall 2017 -- Lecture 9

  10. Example continued: PHY 711 Fall 2017 -- Lecture 9

  11. Another example: PHY 711 Fall 2017 -- Lecture 9

  12. 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. q R PHY 711 Fall 2017 -- Lecture 9

  13. Example continued PHY 711 Fall 2017 -- Lecture 9

  14. Example continued PHY 711 Fall 2017 -- Lecture 9

  15. Consider a particle of mass m moving frictionlessly on a parabola z=c(x2+y2) under the influence of gravity. Find the equations of motion, particularly showing stable circular motion. PHY 711 Fall 2017 -- Lecture 9

  16. PHY 711 Fall 2017 -- Lecture 9

  17. Stable solution when these terms add to 0: PHY 711 Fall 2017 -- Lecture 9

  18. Analysis of stable (circular) motion PHY 711 Fall 2017 -- Lecture 9

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