1. Variational Principles and Lagrange’s Equations

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

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

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. Recipe Best Fit 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:
Structure
Physical Laws
Best Fit
Structure
Fields

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. Recipe Best Fit 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:
Structure
Physical Laws
Best Fit
Structure
Fields

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

26. Recipe Best Fit 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:
Structure
Physical Laws
Best Fit
Structure
Fields

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. 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
Sir Isaac Newton
(1643 – 1727)

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. Electromagnetic field
Lorentz force! 
Hendrik Lorentz
( )

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. 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
Structure
Physical Laws
Best Fit

39. 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
Pierre-Louis Moreau
de Maupertuis
(1698 – 1759)

40. 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 ...”
Richard Phillips
Feynman
(1918 – 1988)

41. 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.”
Freeman John Dyson
(born 1923)

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:

51. Cylindrically symmetric potential
Motion in a potential that depends only on the distance to the z axis
It is convenient to work in cylindrical coordinates
Then

52. Cylindrically symmetric potential
How to rewrite the equations of motion in cylindrical coordinates?

53. Generalized coordinates
Instead of re-deriving the Euler-Lagrange equations explicitly for each problem (e.g. cylindrical coordinates), we introduce a concept of generalized coordinates
Let us consider a set of coordinates
Assume that the Euler-Lagrange equations hold for these variables
Consider a new set of (generalized) coordinates

54. Generalized coordinates
We can, in theory, invert these equations:
Let us do some calculations:

55. Generalized coordinates
The Euler-Lagrange equations are the same in generalized coordinates!!!

56. Generalized coordinates
If the Euler-Lagrange equations are true for one set of coordinates, then they are also true for the other set

57. Cylindrically symmetric potential
Radial force causes a change in radial momentum and a centripetal acceleration

58. Cylindrically symmetric potential
Angular momentum relative to the z axis is a constant

59. Cylindrically symmetric potential
Axial component of velocity does not change

60. Symmetries and conservation laws
The most beautiful and useful illustration of the “structure vs observed behavior” philosophy is the link between symmetries and conservation laws
Conjugate momentum for coordinate :
If Lagrangian does not depend on a certain coordinate, this coordinate is called cyclic (ignorable)
For cyclic coordinates, conjugate momenta are conserved

61. Symmetries and conservation laws
For cyclic coordinates, conjugate momenta are conserved
p = const
p ≠ const

62. Cylindrically symmetric potential
Cyclic coordinates:
Rotational symmetry Translational symmetry
Conjugate momenta:

63. Electromagnetism
Conjugate momenta:

64. Noether’s theorem
Relationship between Lagrangian symmetries and conserved quantities was formalized only in 1915 by Emmy Noether:
“For each symmetry of the Lagrangian, there is a conserved quantity”
Let the Lagrangian be invariant under
the change of coordinates:
α is a small parameter. This invariance
has to hold to the first order in α
Emmy Noether/
Amalie Nöther
(1882 – 1935)

65. Noether’s theorem Invariance of the Lagrangian:
Using the Euler-Lagrange equations

66. Example
Motion in an x-y plane of a mass on a spring (zero equilibrium length):
The Lagrangian is invariant (to the first order in α) under the following change of coordinates:
Then, from Noether’s theorem it follows that

67. Example In polar coordinates: The conserved quantity:
Angular momentum in the x-y plane is conserved

68. Example
For the same problem, we can start with a Lagrangian expressed in polar coordinates:
The Lagrangian is invariant (to any order in α) under the following change of coordinates:
The conserved quantity from Noether’s theorem:

69. 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

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

71. More on symmetries Full time derivative of a Lagrangian:
From the Euler-Lagrange equations: If

72. What is H? Let us expand the Lagrangian in powers of :
Form calculus, for a homogeneous function f of degree n (Euler’s theorem) :

73. What is H?
If the Lagrangian has a form:
Then
For electromagnetism:

74. Conservation of energy
In the field formalism, the conservation of H is a part of Noether’s theorem

75. The brachistochrone problem
Similarly to the “H-trick”:
!!!
Scary! 

76. The brachistochrone problem
Change of variables:
Parametric solution
(cycloid)

77. Scale invariance For Lagrangians of the following form:
And homogeneous L0 of degree k
Introducing scale and time transformations
Then

78. Scale invariance Therefore, after transformations If Then
The Euler-Lagrange equations after transformations
The same!

79. Scale invariance
So, the Euler-Lagrange equations after transformations are the same if
Free fall
Let us recall

80. Scale invariance
So, the Euler-Lagrange equations after transformations are the same if
Mass on a spring
Let us recall

81. Scale invariance
So, the Euler-Lagrange equations after transformations are the same if
Kepler’s problem
Let us recall 3rd Kepler’s law
Johannes Kepler
( )

82. How about open systems?
For some systems we can neglect their interaction with the outside world and formulate their behavior in terms of Lagrangian formalism
For some systems we can not do it
Approach: to describe the system without “leaks” and “feeds” and then add them to the description of the system

83. How about open systems?
For open systems, we first describe the system without “leaks” and “feeds”
After that we add “leaks” and “feeds” to the description of the system
Q: Non-conservative generalized forces

84. Generalized forces Forces 1: Conservative (Potential)
2: Non-conservative

85. Richard Phillips
Feynman
(1918 – 1988)
Generalized forces
In principle, there is no need to introduce generalized forces for a closed system fully described by a Lagrangian
Feynman: “…The principle of least action
only works for conservative systems —
where all forces can be gotten from a potential function. … On a microscopic level — on the deepest level of physics — there are no non-conservative forces. Non-conservative forces, like friction, appear only because we neglect microscopic complications — there are just too many particles to analyze.”
So, introduction of non-conservative forces is a result of the open-system approach

86. Degrees of freedom
The number of degrees of freedom is the number of independent coordinates that must be specified in order to define uniquely the state of the system
For a system of N free particle there are 3N degrees of freedom (3N coordinates) N

87. N k Constraints We can imposed k constraints on the system
The number of degrees of freedom is reduced to 3N – k = s
It is convenient to think of the remaining s independent coordinates as the coordinates of a single point in an s-dimensional space: configuration space N k

88. Types of constraints
Holonomic (integrable) constraints can be expressed in the form:
Nonholonomic constraints cannot be expressed in this form
Rheonomous constraints – contain time dependence explicitly
Scleronomous constraints – do not contain time dependence explicitly

89. Analysis of systems with holonomic constraints
Elimination of variables using constraints equations
Use of independent generalized coordinates
Lagrange’s multiplier method

90. Double 2D pendulum An example of a holonomic scleronomous constraint
The trajectories of the system are very complex
Lagrangian approach produces equation of motion
We need 2 independent generalized coordinates
(N = 2, k = 2 + 2, s = 3 N – k = 2)

91. Double 2D pendulum Relative to the pivot, the Cartesian coordinates
Taking the time derivative, and then squaring
Lagrangian in Cartesian coordinates:

92. Double 2D pendulum Lagrangian in new coordinates:
The equations of motion:

93. Double 2D pendulum Special case The equations of motion: More fun at:

94. Lagrange’s multiplier method
Used when constraint reactions are the object of interest
Instead of considering 3N - k variables and equations, this method deals with 3N + k variables
As a results, we obtain 3N trajectories and k constraint reactions
Lagrange’s multiplier method can be applied to some nonholonomic constraints

95. Lagrange’s multiplier method
Let us explicitly incorporate constraints into the structure of our system
For observable trajectories So

96. Lagrange’s multiplier method
- constraint reactions
Now we have 3N + k equations for and

97. Application to a nonholonomic case
A particle on a smooth hemisphere
One nonholonomic constraint:
While the particle remains on the sphere, the constraint is holonomic
And the reaction from the surface is not zero

98. Application to a nonholonomic case
Constraint equation in cylindrical coordinates:
New Lagrangian in cylindrical coordinates:
Equations of motion

99. Application to a nonholonomic case
Constraint equation in cylindrical coordinates:
New Lagrangian in cylindrical coordinates:
Equations of motion

100. Application to a nonholonomic case
Constraint equation in cylindrical coordinates:
New Lagrangian in cylindrical coordinates:
Equations of motion
Trivial

101. Application to a nonholonomic case
Constraint reaction:

102. Application to a nonholonomic case
Constraint reaction:
Reaction disappears when
The particle becomes airborne