# 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

P

i

=

L

Q
˙

i

.

{\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. ### 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. ### I Proof that canonical transformation implies symplectic condition

Goldstein's Classical Mechanics makes the claim (pages 382 to 383) that given coordinates ##q,p##, Hamiltonian ##H##, and new coordinates ##Q(q,p),P(q,p)##, there exists a transformed Hamiltonian ##K## such that ##\dot Q_i = \frac{\partial K}{ \partial P_i}## and ##\dot P_i = -\frac{\partial...
3. ### Time dependent canonical transformation

THe question is pretty simple. I was doing an exercise, in which $$p = \lambda P, Q = \lambda q$$ is a canonical transformation. We can check it by $$\{Q,P \} = 1$$ But, if we add $$t' = \lambda ^2 t$$, the question says that the transformation is not canonical anymore. I am a little...
4. ### I Canonical transformation vs symplectomorphism

I have read that canonical transformation is basically a symplectomorphism which leaves the symplectic form invariant. My understanding is that the canonical transformation is a passive picture where we keep the point on the phase space fixed and change the coordinate chart, where...
5. ### Change in Hamiltonian under Non-Canonical transformation

I know that if the transformation was canonical, the form of Hamilton's equation would remain invariant. If the generating function for the transformation was time independent, then the Hamiltonian would be invariant and we could directly replace q and p with the transformation equations to get...
6. ### Phase space of a harmonic oscillator and a pendulum

Hello everybody, new here. Sorry in advance if I didn't follow a specific guideline to ask this. Anyways, I've got as a homework assignment two cannonical transformations (q,p)-->(Q,P). I have to obtain the hamiltonian of a harmonic oscillator, and then the new coordinates and the hamiltonian...
7. ### Canonical transformation in classical mechanics

I'm stuck from the beginning. I though I understood the difference between ## \delta## and ##d##, but apparently I was wrong, because I don't know how to exploit it here... Any hint would be greatly appreciated Thank Ric
8. ### Canonical invariance vs. Lorentz invariance

Homework Statement I have an assignment to prove that specific intensity over frequency cubed is Lorentz invariant. One of the main tasks there is to prove the invariance of phase space d^3q \ d^3p and I am trying to prove it with symplectic geometry. I am following Jorge V. Jose and Eugene J...
9. ### How can I improve the legibility of my homework post for better responses?

Homework Statement [/B]Homework EquationsThe Attempt at a Solution [/B] From Poisson bracket relation I have arrived at this point Can anyone please suggest to proceed further
10. ### A Canonical transformation - derviation problem

Let me show you part of a book "Mechanics From Newton’s Laws to Deterministic Chaos" by Florian Scheck. I do not understand why these integrands can differ by more than time derivative of some function M. Why doesn't it change the value of integrals? It seems this point is crucial for me to...
11. ### How can I use Poisson bracket to find P in a canonical transformation?

Homework Statement q,p transforms canonicaly to Q,P where given Q=q(t+s)+(t+s)p ,t is time and s is constt To find P Homework Equations Poisson bracket {Q,P}qp=1 The Attempt at a Solution Using Poisson bracket I find (t+s)*(dP/dp-dP/dq)=1

17. ### 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. ### 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. ### Canonical Transformation (two degrees of freedom)

Homework Statement Point transformation in a system with 2 degrees of freedom is: $$Q_1=q_1^2\\Q_2=q_q+q_2$$ a) find the most general $P_1$ and $P_2$ such that overall transformation is canonical b) Show that for some $P_1$ and $P_2$ the hamiltonain...
20. ### Finding canonical transformation

Homework Statement If in a system with i degrees of freedom the $$Q_i$$ are given what is the best way to proceed for finding the $$P_i$$ so that we have an overall canonical transformation. say for a two degree freedom system we have $$Q_1=q_1^2$$ and $$Q_2=q_1+q_2$$ Homework Equations...
21. ### Fundamental Poisson Bracket - Canonical Transformation

When proofing the poisson brackets algebraically, what is the tool of choice. Can one use the muti dimensionale chain rule or how is it usally done?
22. ### Finding the generator of a transformation

Homework Statement Consider ##\mathscr{H} = \frac12 p^2 + \frac12 x^2, ## which is invariant under infinitesimal rotations in phase space ( the ##x-p## plane). Find the generator of this transformation (after verifying that it is canonical). Homework EquationsThe Attempt at a Solution So the...
23. ### Showing infinitesimal transformation is canonical

Homework Statement So we have infinitesimal transformations from ##q_i## to ##\bar{q_i}## and ##p_i## to ##\bar{p_i}## ( where ##p_i## represents the canonical momentum conjugate of ##q_i##) given by $$\bar{q_i} = q_i + \epsilon \frac{\partial g}{\partial p_i}$$ \bar{p_i} = p_i - \epsilon...
24. ### Non-canonical form into canonical transformation 1-d partial dif.

Homework Statement Problem 29. Use the subtraction trick U(tilda) = U−U1 to reduce the following problems with non-canonical boundary conditions to the canonical ones and write down the equations in terms of the variable ˜u (do not solve them). Note that there are infinitely many u1’s that...
25. ### Canonical Transformation / Poisson Brackets

Question: (A) Show that the following transformation is a canonical transformation: Q = p + aq P = (p - aq)/(2a) (B) Find a generating functions for this transformation. Attempt of Solution: Alright, so this seems to be a very straight forward problem. Transformations are canonical...
26. ### Given a canonical transformation, how does one find its type?

I'm given the following transformation X=x \cos \alpha - \frac{p_y}{\beta} \sin \alpha Y=y \cos \alpha - \frac{p_x}{\beta} \sin \alpha P_X=\beta y \sin \alpha + p_x \cos \alpha P_Y=\beta x \sin \alpha + p_y \cos \alpha and I'm asked to find what type(s) of transformation it is. I'm not...
27. ### Canonical transformation problem

Homework Statement Let Q^1 = (q^1)^2, Q^2 = q^1+q^2, P_{\alpha} = P_{\alpha}\left(q,p \right), \alpha = 1,2 be a CT in two freedoms. (a) Complete the transformation by finding the most general expression for the P_{\alpha}. (b) Find a particular choice for the P_{\alpha} that will reduce the...
28. ### Question about canonical transformation

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. ### Verifying a Canonical Transformation with Poisson Brackets

Homework Statement Show that Q_{1}=\frac{1}{\sqrt{2}}(q_{1}+\frac{p_{2}}{mω}) Q_{2}=\frac{1}{\sqrt{2}}(q_{1}-\frac{p_{2}}{mω}) P_{1}=\frac{1}{\sqrt{2}}(p_{1}-mωq_{2}) P_{2}=\frac{1}{\sqrt{2}}(p_{1}+mωq_{2}) (where mω is a constant) is a canonical transformation by Poisson bracket test. This...
30. ### Infinitesimal Canonical Transformation

Hi all! Another questions which is due to the gaps in my calculus knowledge. In these notes: http://people.hofstra.edu/Gregory_C_Levine/qft.pdf in the line above eq. (1) where it says that notation P is now unecessary, is it because \partial{ (p+\delta p)} is much smaller than p+\delta p...
31. ### Understanding the Concept of Canonical Transformation in Hamiltonian Mechanics

There's a part in my book that I don't understand. I have attached the part and it is basically about how to transform from a set of conjugate variables (q,p) to another (Q,P) while preserving the hamilton equations of motion. I don't understand what he means by q,Q being separately independent...
32. ### Canonical transformation for Harmonic oscillator

Find under what conditions the transformation from (x,p) to (Q,P) is canonical when the transformation equations are: Q = ap/x , P=bx2 And apply the transformation to the harmonic oscillator. I did the first part and found a = -1/2b I am unsure about the next part tho: We have the...
33. ### Show condition for canonical transformation

Homework Statement Consider the transformation from the variables (q,p) to (Q,P) by virtue of q = q(Q,P), p = p(Q,P) and H(q,p,t) = H(Q,P,t). Show that the equations of motion for Q,P are: \partialH/\partialQ = -JDdP/dt \partialH/\partialP = JDdQ/dt where JD is the Jacobian determinant...
34. ### Exploring Canonical Transformations

I have posted before this, an example in which I struggled through. Now am gnna ask something more general, for me and for the students who suffer from studying a material alone. If you were asked to prove that the time-independent transformation P=.. and Q=.. is canonical. And finding the...
35. ### Hamiltonian, generating function, canonical transformation

Homework Statement Consider a harmonic oscillator with generalized coordinates q and p with a frequency omega and mass m. Let the transformation (p,q) -> (Q,P) be such that F_2(q,P,t)=\frac{qP}{\cos \theta }-\frac{m\omega }{2}(q^2+P^2)\tan \theta. 1)Find K(Q,P) where \theta is a function of...
36. ### Finding a generating function for a canonical transformation

Homework Statement I'm trying to find a generating function for the canonical transformation Q=\left ( \frac{\sin p}{q} \right ), P=q \cot p.Homework Equations I am not really sure. I know there are 4 different types of generating function. I guess it's totally up to me to choose the type of...
37. ### Canonical transformation between two given hamiltonians

Hello everyone, I am given the inital hamiltonian H = (1/2)*(px2x4 - 2iypy + 1/x2) and the transformed hamiltonian K = (1/2)*(Px2 + Py2 + X2 + Y2) and I'm supposed to show there exists a canonical transformation that transforms H to K and find it. I don't know how to solve problems of this sort...
38. ### Canonical Transformation of the Hubbard Model

Hi, Suppose we have a 2 site Hubbard model, with the hopping Hamiltonian given by H_t and the Coulomb interaction Hamiltonian given by \hat{H}_U. In the strong coupling limit (U/t >> 1), we define a canonical transformation of \hat{H} = \hat{H}_U + \hat{H}_t, as H' =...
39. ### Canonical transformation in Hamiltonian

Hamiltonian H=\frac{1}{2m}(P+\frac{e}{c}A)^{2} - e\phi and H^{'}=\frac{1}{2m}(P+\frac{e}{c}A^{'})^{2} - e\phi^{'} With gauge: A^{'}=A+\nabla\chi and \phi^{'}=\phi-\frac{1}{c}\dot{\chi} Why H^{'}-\frac{e}{c}\dot{\chi}=e^{-\frac{ie\chi}{\hbar c}}He^{\frac{ie\chi}{\hbar c}} ? Thanks.
40. ### 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. ### 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. ### 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. ### 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. ### How Can I Verify a Canonical Transformation Using the Poisson Bracket?

Homework Statement Verify that q_bar=ln(q^-1*sin(p)) p_bar=q*cot(p) * represents muliplication sorry i don't know how to use the programming to make it look better 2. The attempt at a solution my problem is that i really have no clue what is going on. I have read...
45. ### Canonical Transformation Problem

Homework Statement Show that the time reversal transformation given by Q = q, P = − p and T = − t, is canonical, in the sense that the form of the Hamiltonian equations of motion is preserved. However, it does not satisfy the invariance of the fundamental Poisson Bracket relations. This is...
46. ### 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. ### 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. ### Canonical Transformation of Parabolic PDEs

Hi again I am studying PDEs and came across a solved problem in my textbook, which describes the transformation of a parabolic second order PDE to canonical form. I want to know how to find the second canonical substitution when one has been computed from the characteristic equation...
49. ### Extended Canonical Transformation

Homework Statement The transformation equations are: Q=q^\alpha cos(\beta p) P=q^\alpha sin(\beta p) For what values of \alpha and \beta do these equations represent an extended canonical transformation? The Attempt at a Solution Well, just for a start, what is the condition for a...
50. ### 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...