Variational Principles and Lagrange’s Equations

1. Variational Principles and Lagrange’s Equations

2. Joseph Louis Lagrange/ Giuseppe Luigi Lagrangia (1736 – 1813) • Definitions • Lagrangian density: • Lagrangian: • Action: • How to find the special value for action corresponding to observable ?

3. Pierre-Louis Moreau de Maupertuis (1698 – 1759) Sir William Rowan Hamilton (1805 – 1865) Richard Phillips Feynman (1918 – 1988) • Variational principle • Maupertuis: Least Action Principle • Hamilton: Hamilton’s Variational Principle • Feynman: Quantum-Mechanical Path Integral Approach

4. Functionals • Functional: given any function f(x), produces a number S • Action is a functional: • Examples of finding special values of functionals using variational approach: • shortest distance between two points on a plane; • the brachistochrone problem; • minimum surface of revolution; • etc.

5. Shortest distance between two points on a plane • An element of length on a plane is • Total length of any curve going between points 1 and 2 is • The condition that the curve is the shortest path is that the functional I takes its minimum value

6. The brachistochrone problem • Find a curve joining two points, along which a particle falling from rest under the influence of gravity travels from the highest to the lowest point in the least time • Brachistochrone solution: the value of the functional t [y(x)] takes its minimum value

7. Calculus of variations • Consider a functional of the following type • What function y(x) yields a stationary value (minimum, maximum, or saddle) of J ?

8. Calculus of variations • Assume that function y0(x) yields a stationary value and consider all possible functions in the form:

9. Calculus of variations • In this case our functional becomes a function of α: • Stationary value condition:

10. Stationary value 1 2 3

11. Stationary value 1 2 3 u dv u v v du

12. Stationary value 1 2 3

13. Stationary value 1 2 3

14. Stationary value arbitrary Trivial … 

15. Stationary value arbitrary Nontrivial !!! 

16. Shortest distance between two points on a plane Straight line! 

17. The brachistochrone problem Scary! 

18. Fields Physical Laws Best Fit Fields Structure Structure • Recipe • 1. Bring together structure and fields • 2. Relate this togetherness to the entire system • 3. Make them fit best when the fields have observable dependencies:

19. Back to trajectories and Lagrangians • How to find the special values for action corresponding to observable trajectories ? • We look for a stationary action using variational principle

20. Fields Physical Laws Best Fit Fields Structure Structure • Recipe • 1. Bring together structure and fields • 2. Relate this togetherness to the entire system • 3. Make them fit best when the fields have observable dependencies:

21. Back to trajectories and Lagrangians • For open systems, we cannot apply variational principle in a consistent way, since integration in not well defined for them • We look for a stationary action using variational principle for closed systems:

22. Stationary value Nontrivial !!! 

23. Simplest non-trivial case • Let’s start with the simplest non-trivial result of the variational calculus and see if it can yield observable trajectories

24. Stationary value Nontrivial !!! 

25. Joseph Louis Lagrange (1736 – 1813) Leonhard Euler (1707 – 1783) • Euler- Lagrange equations • These equations are called the Euler- Lagrange equations

26. Fields Physical Laws Best Fit Fields Structure Structure • Recipe • 1. Bring together structure and fields • 2. Relate this togetherness to the entire system • 3. Make them fit best when the fields have observable dependencies:

27. How to construct Lagrangians? • Let us recall some kindergarten stuff • On our – classical-mechanical – level, we know several types of fundamental interactions: • Gravitational • Electromagnetic • That’s it

28. Sir Isaac Newton (1643 – 1727) • Gravitation • For a particle in a gravitational field, the trajectory is described via 2nd Newton’s Law: • This system can be approximated as closed • The structure (symmetry) of the system is described by the gravitational potential

29. Electromagnetic field • For a charged particle in an electromagnetic field, the trajectory is described via 2nd Newton’s Law: • This system can be approximated as closed • The structure (symmetry) of the system is described by the scalar and vector potentials Really???

30. Electromagnetic field

31. Electromagnetic field

32. Hendrik Lorentz (1853-1928) • Electromagnetic field • Lorentz force! 

33. Kindergarten • Thereby: • In component form

34. How to construct Lagrangians? • Kindergarten stuff: • The “kindergarten equations” look very similar to the Euler-Lagrange equations! We may be on the right track! 

35. Gravitation

36. Gravitation

37. Electromagnetism

38. Physical Laws Best Fit Structure • Bottom line • We successfully demonstrated applicability of our recipe • This approach works not just in classical mechanics only, but in all other fields of physics

39. Pierre-Louis Moreau de Maupertuis (1698 – 1759) • Some philosophy • de Maupertuis on the principle of least action (“Essai de cosmologie”, 1750): “In all the changes that take place in the universe, the sum of the products of each body multiplied by the distance it moves and by the speed with which it moves is the least that is possible.” • How does an object know in advance • what trajectory corresponds to a • stationary action??? • Answer: quantum-mechanical path • integral approach

40. Richard Phillips Feynman (1918 – 1988) • Some philosophy • Feynman: “Is it true that the particle doesn't just "take the right path" but that it looks at all the other possible trajectories? ... The miracle of it all is, of course, that it does just that. ... It isn't that a particle takes the path of least action but that it smells all the paths in the neighborhood and chooses the one that has the least action ...”

41. Freeman John Dyson (born 1923) • Some philosophy • Dyson: “In 1949, Dick Feynman told me about his "sum over histories" version of quantum mechanics. "The electron does anything it likes," he said. "It just goes in any direction at any speed, forward or backward in time, however it likes, and then you add up the amplitudes and it gives you the wave-function." I said to him, "You're crazy." But he wasn't.”

42. Some philosophy • Philosophical meaning of the Lagrangian formalism: structure of a system determines its observable behavior • So, that's it? • Why do we need all this? • In addition to the deep philosophical meaning, Lagrangian formalism offers great many advantages compared to the Newtonian approach

43. Lagrangian approach: extra goodies • It is scalar (Newtonian – vectorial) • Allows introduction of configuration space and efficient description of systems with constrains • Becomes relatively simpler as the mechanical system becomes more complex • Applicable outside Newtonian mechanics • Relates conservation laws with symmetries • Scale invariance applications • Gauge invariance applications

44. Simple example • Projectile motion

45. Another example • Another Lagrangian • What is going on?!

46. Gauge invariance • For the Lagrangians of the type • And functions of the type • Let’s introduce a transformation (gauge transformation):

47. Gauge invariance

48. Gauge invariance

49. Gauge invariance

50. Back to the question: How to construct Lagrangians? • Ambiguity: different Lagrangians result in the same equations of motion • How to select a Lagrangian appropriately? • It is a matter of taste and art • It is a question of symmetries of the physical system one wishes to describe • Conventionally, and for expediency, for most applications in classical mechanics: