1. Hamiltonian Formalism

2. Legendre transformations
Adrien-Marie
Legendre
(1752 –1833)

3. What is H?
Conjugate momentum
Then So

4. What is H?

5. What is H?

6. What is H? If
Then
Kinetic energy
In generalized coordinates

7. What is H?
For scleronomous generalized coordinates
Then If

8. What is H?
For scleronomous generalized coordinates, H is a total mechanical energy of the system (even if H depends explicitly on time)
If H does not depend explicitly on time, it is a constant of motion (even if is not a total mechanical energy)
In all other cases, H is neither a total mechanical energy, nor a constant of motion

9. Hamilton’s equations Hamiltonian: Hamilton’s equations of motion:
Sir William Rowan
Hamilton
(1805 – 1865)

10. Hamiltonian formalism
For a system with M degrees of freedom, we have 2M independent variables q and p: 2M-dimensional phase space (vs. configuration space in Lagrangian formalism)
Instead of M second-order differential equations in the Lagrangian formalism we work with 2M first-order differential equations in the Hamiltonian formalism
Hamiltonian approach works best for closed holonomic systems
Hamiltonian approach is particularly useful in quantum mechanics, statistical physics, nonlinear physics, perturbation theory

11. Hamiltonian formalism for open systems

12. Hamilton’s equations in symplectic notation
Construct a column matrix (vector) with 2M elements
Then
Construct a 2Mx2M square matrix as follows:

13. Hamilton’s equations in symplectic notation
Then the equations of motion will look compact in the symplectic (matrix) notation:
Example (M = 2):

14. Lagrangian to Hamiltonian
Obtain conjugate momenta from a Lagrangian
Write a Hamiltonian
Obtain from
Plug into the Hamiltonian to make it a function of coordinates, momenta, and time

15. Lagrangian to Hamiltonian
For a Lagrangian quadratic in generalized velocities
Write a symplectic notation:
Then a Hamiltonian
Conjugate momenta

16. Lagrangian to Hamiltonian
Inverting this equation
Then a Hamiltonian

17. Example: electromagnetism

18. Example: electromagnetism

19. Hamilton’s equations from the variational principle
Action functional :
Variations in the phase space :

20. Hamilton’s equations from the variational principle
Integrating by parts

21. Hamilton’s equations from the variational principle
For arbitrary independent variations

22. Conservation laws
If a Hamiltonian does not depend on a certain coordinate explicitly (cyclic), the corresponding conjugate momentum is a constant of motion
If a Hamiltonian does not depend on a certain conjugate momentum explicitly (cyclic), the corresponding coordinate is a constant of motion
If a Hamiltonian does not depend on time explicitly, this Hamiltonian is a constant of motion

23. Higher-derivative Lagrangians
Let us recall:
Lagrangians with i > 1 occur in many systems and theories:
Non-relativistic classical radiating charged particle (see Jackson)
Dirac’s relativistic generalization of that
Nonlinear dynamics
Cosmology
String theory
Etc.

24. Higher-derivative Lagrangians
For simplicity, consider a 1D case:
Variation
Mikhail Vasilievich Ostrogradsky ( )

25. Higher-derivative Lagrangians

26. Higher-derivative Lagrangians

27. Higher-derivative Lagrangians
Generalized coordinates/momenta:

28. Higher-derivative Lagrangians
Euler-Lagrange equations:
We have formulated a ‘higher-order’ Lagrangian formalism
What kind of behavior does it produce?

29. Example

30. Example

31. Example
H is conserved and it generates evolution – it is a Hamiltonian!
Hamiltonian linear in momentum?!?!?!
No low boundary on the total energy – lack of ground state!!!
Produces ‘runaway’ solutions: the system becomes highly unstable - collapse and explosion at the same time

32. ‘Runaway’ solutions
Unrestricted low boundary of the total energy produces instabilities
Additionally, we generate new degrees of freedom, which require introduction of additional (originally unknown) initial conditions for them
These problems are solved by means of introduction of constraints
Constraints restrict unstable behavior and eliminate unnecessary new degrees of freedom

33. Canonical transformations
9.1
Canonical transformations
Recall gauge invariance (leaves the evolution of the system unchanged):
Let’s combine gauge invariance with Legendre transformation:
K – is the new Hamiltonian (‘Kamiltonian’ )
K may be functionally different from H

34. Canonical transformations
9.1
Canonical transformations
Multiplying by the time differential: So

35. 9.1
Generating functions
Such functions are called generating functions of canonical transformations
They are functions of both the old and the new canonical variables, so establish a link between the two sets
Legendre transformations may yield a variety of other generating functions

36. Generating functions We have three additional choices:
9.1
Generating functions
We have three additional choices:
Canonical transformations may also be produced by a mixture of the four generating functions

37. An example of a canonical transformation
9.2
An example of a canonical transformation
Generalized coordinates are indistinguishable from their conjugate momenta, and the nomenclature for them is arbitrary
Bottom-line: generalized coordinates and their conjugate momenta should be treated equally in the phase space

38. Criterion for canonical transformations
9.4
Criterion for canonical transformations
How to make sure this transformation is canonical?
On the other hand If
Then

39. Criterion for canonical transformations
9.4
Criterion for canonical transformations
Similarly, If
Then

40. Criterion for canonical transformations
9.4
Criterion for canonical transformations
So, If

41. Canonical transformations in a symplectic form
9.4
Canonical transformations in a symplectic form
After transformation
On the other hand

42. Canonical transformations in a symplectic form
9.4
Canonical transformations in a symplectic form
For the transformations to be canonical:
Hence, the canonicity criterion is:
For the case M = 1, it is reduced to (check yourself)

43. 1D harmonic oscillator Let us find a conserved canonical momentum
9.3
1D harmonic oscillator
Let us find a conserved canonical momentum
Generating function

44. 1D harmonic oscillator Nonlinear partial differential equation for F 
9.3
1D harmonic oscillator
Nonlinear partial differential equation for F 
Let’s try to separate variables
Let’s try

45. 9.3
1D harmonic oscillator
We found a generating function!

46. 9.3
1D harmonic oscillator

47. 9.3
1D harmonic oscillator

48. 9.5
Canonical invariants
What remains invariant after a canonical transformation?
Matrix A is a Jacobian of a space transformation
From calculus, for elementary volumes:
Transformation is canonical if

49. Canonical invariants For a volume in the phase space
9.5
Canonical invariants
For a volume in the phase space
Magnitude of volume in the phase space is invariant with respect to canonical transformations:

50. 9.5
Canonical invariants
What else remains invariant after canonical transformations?

51. 9.5
Canonical invariants
For M = 1
For many variables

52. Poisson brackets Poisson brackets:
9.5
Poisson brackets
Poisson brackets:
Poisson brackets are invariant with respect to any canonical transformation
Siméon Denis Poisson
(1781 – 1840)

53. 9.5
Poisson brackets
Properties of Poisson brackets :

54. Poisson brackets In matrix element notation:
9.5
Poisson brackets
In matrix element notation:
In quantum mechanics, for the commutators of coordinate and momentum operators:

55. Poisson brackets and equations of motion
9.6
Poisson brackets and equations of motion

56. Poisson brackets and conservation laws
9.6
Poisson brackets and conservation laws
If u is a constant of motion
If u has no explicit time dependence
In quantum mechanics, conserved quantities commute with the Hamiltonian

57. Poisson brackets and conservation laws
9.6
Poisson brackets and conservation laws
If u and v are constants of motion with no explicit time dependence
For Poisson brackets:
If we know at least two constants of motion, we can obtain further constants of motion

58. Infinitesimal canonical transformations
9.4
Infinitesimal canonical transformations
Let us consider a canonical transformation with the following generating function (ε – small parameter):
Then

59. Infinitesimal canonical transformations
9.4
Infinitesimal canonical transformations
Multiplying by dt
Then

60. Infinitesimal canonical transformations
9.4
Infinitesimal canonical transformations
Infinitesimal canonical transformations:
In symplectic notation:

61. 9.6
Evolution generation
Motion of the system in time interval dt can be described as an infinitesimal transformation generated by the Hamiltonian
The system motion in a finite time interval is a succession of infinitesimal transformations, equivalent to a single finite canonical transformation
Evolution of the system is a canonical transformation!!!

62. Application to statistical mechanics
9.9
Application to statistical mechanics
In statistical mechanics we deal with huge numbers of particles
Instead of describing each particle separately, we describe a given state of the system
Each state of the system represents a point in the phase space
We cannot determine the initial conditions exactly
Instead, we study a certain phase volume – ensemble – as it evolves in time

63. Application to statistical mechanics
9.9
Application to statistical mechanics
Ensemble can be described by its density – a number of representative points in a given phase volume
The number of representative points does not change
Ensemble evolution can be thought as a canonical transformation generated by the Hamiltonian
Volume of a phase space is a constant for a canonical transformation

64. Application to statistical mechanics
9.9
Application to statistical mechanics
Ensemble is evolving so its density is evolving too
On the other hand
Liouville’s theorem
In statistical equilibrium
Joseph Liouville
( )

65. Hamilton–Jacobi theory
10.1
Hamilton–Jacobi theory
We can look for the following canonical transformation, relating the constant (e.g. initial) values of the variables with the current ones:
The reverse transformations will give us a complete solution

66. Hamilton–Jacobi theory
10.1
Hamilton–Jacobi theory
Let us assume that the Kamiltonian is identically zero
Then
Choosing the following generating function
Then, for such canonical transformation:

67. Hamilton–Jacobi theory
10.1
Hamilton–Jacobi theory
Hamilton–Jacobi equation
Conventionally: Hamilton’s principal function
Partial differential equation
First order differential equation
Number of variables: M + 1
Sir William Rowan
Hamilton
(1805 – 1865)
Karl Gustav Jacob
Jacobi
(1804 – 1851)

68. Hamilton–Jacobi theory
10.1
Hamilton–Jacobi theory
Suppose the solution exists, so it will produce M + 1 constants of integration:
One constant is evident:
We chose those M constants to be the new momenta
While the old momenta

69. Hamilton–Jacobi theory
10.1
Hamilton–Jacobi theory
We relate the constants with the initial values of our old variables:
The new coordinates are defined as:
Inverting those formulas we solve our problem

70. 10.1
Have we met before?
Remember action?

71. Hamilton’s characteristic function
10.1
Hamilton’s characteristic function
When the Hamiltonian does not depend on time explicitly
Generating function (Hamilton’s characteristic function)

72. Hamilton’s characteristic function
10.3
Hamilton’s characteristic function
Now we require:
So:
Detailed comparison of Hamilton’s characteristic vs. Hamilton’s principal is given in a textbook (10.3)

73. Hamilton’s characteristic function
10.3
Hamilton’s characteristic function
What is the relationship between S and W ?
One of possible relationships (the most conventional):

74. 10.6
Periodic motion
For energies small enough we have periodic oscillations (librations) – green curves
For energies great enough we msy have periodic rotations – red curves
Blue curve – separatrix trajectory – bifurcation transition from librations to rotations

75. Action-angle variables
10.6
Action-angle variables
For either type of periodic motion let us introduce a new variable – action variable (don’t confuse with action!):
A generalized coordinate conjugate to action variable is the angle variable:
The equation of motion for the angle variable:

76. Action-angle variables
10.6
Action-angle variables
In a compete cycle
This is a frequency of the periodic motion

77. Example: 1D Harmonic oscillator
10.2
Example: 1D Harmonic oscillator

78. Example: 1D Harmonic oscillator
10.2
Example: 1D Harmonic oscillator

79. Action-angle variables for 1D harmonic oscillator
10.6
Action-angle variables for 1D harmonic oscillator
Therefore, for the frequency:

80. Separation of variables in the Hamilton-Jacobi equation
10.4
Separation of variables in the Hamilton-Jacobi equation
Sometimes, the principal function can be successfully separated in the following way:
For the Hamiltonian without an explicit time dependence:
Functions Hi may or may not be Hamiltonians