1 / 80

Hamiltonian Formalism

Hamiltonian Formalism. Adrien-Marie Legendre (1752 –1833). Legendre transformations Legendre transformation :. What is H? Conjugate momentum Then So. What is H?. What is H?. What is H? If Then Kinetic energy In generalized coordinates. What is H?

hancockj
Download Presentation

Hamiltonian Formalism

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Hamiltonian Formalism

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

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

  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. Mikhail Vasilievich Ostrogradsky (1801 - 1862) • Higher-derivative Lagrangians • For simplicity, consider a 1D case: • Variation

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

  37. 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. 9.4 • Criterion for canonical transformations • How to make sure this transformation is canonical? • On the other hand • If • Then

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

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

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

  42. 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. 9.3 • 1D harmonic oscillator • Let us find a conserved canonical momentum • Generating function

  44. 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. 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?

More Related