What is Canonical transformation: Definition and 55 Discussions

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (q, p, t) → (Q, P, t) that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi equations (a useful method for calculating conserved quantities) and Liouville's theorem (itself the basis for classical statistical mechanics).
Since Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates q → Q do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if we simultaneously change the momentum by a Legendre transformation into








{\displaystyle P_{i}={\frac {\partial L}{\partial {\dot {Q}}_{i}}}.}
Therefore, coordinate transformations (also called point transformations) are a type of canonical transformation. However, the class of canonical transformations is much broader, since the old generalized coordinates, momenta and even time may be combined to form the new generalized coordinates and momenta. Canonical transformations that do not include the time explicitly are called restricted canonical transformations (many textbooks consider only this type).
For clarity, we restrict the presentation here to calculus and classical mechanics. Readers familiar with more advanced mathematics such as cotangent bundles, exterior derivatives and symplectic manifolds should read the related symplectomorphism article. (Canonical transformations are a special case of a symplectomorphism.) However, a brief introduction to the modern mathematical description is included at the end of this article.

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  1. A

    I Does the Hamilton-Jacobi equation exist for chaotic systems?

    Given a Hamiltonian ##H(\mathbf{q},\mathbf{p})##, in the time-independent Hamilton-Jacobi approach we look for a canonical transformation ##(\mathbf{q},\mathbf{p})\rightarrow(\mathbf{Q},\mathbf{P})## such that the new Hamiltonian is one of the new momenta...
  2. Lagrange fanboy

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  3. LCSphysicist

    Time dependent canonical transformation

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  4. L

    I Canonical transformation vs symplectomorphism

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  5. Saptarshi Sarkar

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  6. D

    Phase space of a harmonic oscillator and a pendulum

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  7. dRic2

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  8. gasar8

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  9. A

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  10. Vicol

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  11. A

    How can I use Poisson bracket to find P in a canonical transformation?

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  12. thecourtholio

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  13. B

    A Test if 2nd order diff eq. can be derived from a Hamiltonian

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  14. CassiopeiaA

    A Symplectic Condition For Canonical Transformation

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  15. F

    I Canonical transformations and generating functions

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  16. F

    Hamiltonian as the generator of time translations

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  17. S

    The restricted canonical transformation group

    Homework Statement Show that the set of restricted canonical transformation forms a group. Verify this statement once using the invariance of Hamilton's principle under canonical transformation, and again using the symplectic condition. Homework Equations (Invariance of Hamilton's principle...
  18. kolawoletech

    A Most General form of Canonical Transformation

    How do I go about finding the most general form of the canonical transformation of the form Q = f(q) + g(p) P = c[f(q) + h(p)] where f,g and h are differential functions and c is a constant not equal to zero. Where (Q,P) and (q,p) represent the generalised cordinates and conjugate momentum in...
  19. S

    Canonical Transformation (two degrees of freedom)

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  20. S

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  21. R

    Fundamental Poisson Bracket - Canonical Transformation

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  22. D

    Finding the generator of a transformation

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  23. D

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  24. P

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  25. Z

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  26. M

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  27. M

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  28. M

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    I was going through my professor's notes about Canonical transformations. He states that a canonical transformation from (q, p) to (Q, P) is one that if which the original coordinates obey Hamilton's canonical equations than so do the transformed coordinates, albeit for a different Hamiltonian...
  29. darida

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  30. L

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  31. A

    Understanding the Concept of Canonical Transformation in Hamiltonian Mechanics

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  32. A

    Canonical transformation for Harmonic oscillator

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  33. A

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  34. M

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  35. fluidistic

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  36. fluidistic

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  37. C

    Canonical transformation between two given hamiltonians

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  38. maverick280857

    Canonical Transformation of the Hubbard Model

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  39. J

    Canonical transformation in Hamiltonian

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  40. I

    BCS theory by canonical transformation

    I am reading Tinkham's "introduction to superconductivity" 1975 by McGraw-Hill, Inc. Tinkham derives the BCS theory by canonical transformation. At the beginning of the chapter he writes: "We start with the observation that the characteristic BCS pair interaction Hamiltonian will lead to a...
  41. S

    Canonical Transformation and harmonic-oscillator

    Show that the transformation Q = p + iaq , P = (p-iaq)/2ia is canonical and find the generating function. Use the transformation to solve the harmonic-oscillator problem. I was able to determine if the transformation is canonical, and it is. However, when it came to finding the...
  42. L

    Generating function for canonical transformation

    Homework Statement Given the transformation Q = p+iaq, P = \frac{p-iaq}{2ia} Homework Equations find the generating function The Attempt at a Solution As far as I know, one needs to find two independent variables and try to solve. I couldn't find such to variables. I've...
  43. R

    Is a Rotated Frame a Canonical Transformation in Classical Mechanics?

    Homework Statement Verify that the change to a rotated frame is a canonical transformation: \bar{x} = x cos\theta - y sin\theta \bar{y} = x sin \theta + y cos \theta \bar{p_x} = p_x cos \theta - p_y sin\theta \bar{p_y} = p_x sin \theta + p_y cos \theta Where [f,g] = poisson bracket Homework...
  44. S

    How Can I Verify a Canonical Transformation Using the Poisson Bracket?

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  45. N

    Canonical Transformation Problem

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  46. S

    Calculating generator function in canonical transformation

    I'm searching for an example of how to find out generator function for a canonical transformation, when new canonical variables are given in terms of old variables. Any help is greatly appreciated.
  47. strangerep

    Most General Canonical Transformation?

    In classical Hamiltonian mechanics, the concept of a canonical transformation ("CT") preserving the form of Hamilton's eqns is well known. Textbooks (e.g., Goldstein) distinguish "restricted" CTs that just mix the q's and p's (generalized coordinates and generalized momenta respectively)...
  48. maverick280857

    Canonical Transformation of Parabolic PDEs

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  49. M

    Extended Canonical Transformation

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  50. L

    How Does a Canonical Transformation Relate to Hamilton's Equations of Motion?

    Homework Statement Consider a canonical transformation with generating function F_2 (q,P) = qP + \epsilon G_2 (q,P), where \epsilon is a small parameter. Write down the explicit form of the transformation. Neglecting terms of order \epsilon^2 and higher,find a relation between this...