Linked Questions

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Condition for a time-dependent transformation to be a canonical transformation [duplicate]

I'm looking for a sufficient condition to determine if a given transformation is a canonical transformation. I have found two conditions, but they are only valid for the case that the transformation ...
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How are the two definitions of Canonical Transformations related/equivalent? [duplicate]

I am aware of two definitions of canonical transformations which I state below. Definition $1$ We go from old set of $\{q_i,p_i,t\}$ of $2n$ phase space variables to a new set $\{Q(q_i,p_i,t),P(q_i,...
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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 ...
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Is a canonical transformation equivalent to a transformation that preserves volume and orientation?

We have seen the reverse statement: Lioville's Theorem states that canonical transformations preserve volume (and orientation as well). Is the reverse true? If I demand a map from the phase space to ...
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4answers
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Why do we care only about canonical transformations?

In Hamiltonian mechanics we search change of coordinates that leaves the Hamilton equation invariant: these are the canonical transformations. My question is: why we want to leave the equations ...
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2answers
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How do contact transformations differ from canonical transformations?

From Goldstein, 3rd edition, section 9.6, page 399 after equation 9.101: [...] The motion of a system in a time interval $dt$ can be described by an infinitesimal contact transformation generated ...
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1answer
210 views

About the group of canonical transformations and the matrices representing them

Recently I have come to know that for a system with $2n$ dimensional phase space, the set of all canonical transformations form a group ${\rm Sp(2n, R)}$. But in contrast to other Lie groups e.g. ${\...
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741 views

Momentum vector transformation

I am confused about the way momentum vector transforms in the following case: $$q_k \to q_k'= q_k + \epsilon f_k(q)$$ The Jacobian is thus $\Lambda_{ij} = \frac{\partial q'_i}{\partial q_j} \approx \...
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Invariant Phase space volume under canonical transfromation

I have a volume element in phase space: $$ d\omega = \prod _{i=1}^{N}(dq_{i},dp_{i})$$ Now I should show the invariance of this product under canonical transformations. I think first I would have to ...
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1answer
726 views

Can all canonical transformation be obtained through generation function approaches?

The question can be formulated as following: Suppose $$\delta \int_{t_1}^{t_2}{[p\cdot \dot{q} - H(p,q,t) ]dt} = 0$$ $$\delta \int_{t_1}^{t_2}{[P\cdot \dot{Q} - K(P,Q,t) ]dt} = 0$$ in which $$P = P(...
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1answer
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For an infinitesimal transformation in phase space, what functions are allowed for this to be a canonical transformation?

Consider an infinitesimal transformation: $$(q_{i},p_{j}) \quad\longrightarrow \quad(Q_{i},P_{j}) ~=~ \left(q_{i} + \alpha F_{i}(q,p),~p_{j} + \alpha E_{j}(q,p)\right) $$ where $α$ is considered to ...
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1answer
487 views

Poisson brackets' property as necessary and sufficient condition for transformation to be canonical?

I read (Landau, Lifshitz: Mechanics) and then I want to know if conditions (45.10) are sufficient for transformation $p,q \to P,Q$ to be canonical (obviously, they are necessary).
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1answer
857 views

Can all symplectic-form preserving canonical transformations generated by generating functions

This question is related to this fascinating post and this post and this post, but more limited in scope in discussing the practical definition canonical transformations. Canonical transformation ...
3
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1answer
517 views

Do canonical transformations form a group?

In a course on classical mechanics, we barely touched upon canonical transformations via generating functions. Just like Lorentz transformations form a group, I want to know if canonical ...
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2answers
768 views

Use of generating function in canonical transformation

In the theory of Canonical transformations, initially we use the fact that the new and the old system of $(q_i, p_i)$ with the Hamiltonian $H$ satisfy the modified Hamilton's principle. Now here, the ...
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2answers
418 views

Time-independent canonical transformations

Lie's criterion tells us that $(q,p) \to (Q,P)$ is a canonical transformation, for a system with Hamiltonian $H$ and "Kamiltonian" $K$, if and only if the identity $$\sum_k p_k dq_k -Hdt = \sum_k P_k ...
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3answers
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How can we derive from $\{G,H\}=0$ that $G$ generates a transformations which leaves the form of Hamilton's equations unchanged?

In the Hamiltonian formalism, a symmetry is defined as transformation generated by a function $G$ is a symmetry if $$\{G,H\}=0 ,$$ where $H$ denotes the Hamiltonian. On the other hand, a symmetry is ...
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1answer
439 views

Can any symplectomorphism be called a canonical transformation?

I just want to make sure I am thinking clearly about canonical coordinates and transformations in Hamiltonian mechanics. Suppose we have a Hamiltonian system $(M, \omega, H)$ — where $M$ is the ...
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1answer
504 views

Modified Hamilton's Principle overconstraining a system by imposing too many boundary conditions

In Hamiltonian Mechanics, a version of Hamilton's principle is shown to evolve a system according to the same equations of motion as the Lagrangian, and therefore Newtonian formalism. In particular, ...
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1answer
383 views

A question about canonical transformation

I have posted this question in math.stackexchange before with no answer till now. It may be more suitable to post here. There is a problem in Arnold's Mathematical Methods of Classical Mechanics ...
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1answer
597 views

Independent canonical coordinate variables?

In Goldstein's Classical Mechanics (2nd ed.) on section 9-1 page 382, there is a discussion about finding a canonical transformation $(q_i,p_i)\rightarrow (Q_j(q_i,p_i,t),P_j(q_i,p_i,t))$ from a given ...
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1answer
459 views

How to check if a generating function produces an identity transformation without substituting the CT equations in the Hamiltonian?

In chapter 9, Goldstein ($3^{rd}$ ed.) includes a discussion and a few "trivial special cases" of Canonical Transformation which keeps the form of the Hamiltonian unchanged and named it Identity ...
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1answer
329 views

Why time-dependent canonical transformation satisfies symplectic condition?

I am reading Chapter 9 of Goldstein. He proves that any time-independent canonical transformations satisfy symplectic condition. And after that, he shows that if we ignore second order small quantity, ...
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1answer
196 views

Are Poisson brackets preserved during a canonical transformation?

Fix a Hamiltonian $H(q, p, t)$. Definition: A transformation $(q, p, t)\mapsto (Q(q, p, t), P(q, p, t), t)$ is said to be canonical iff for the Kamiltonian $K$ defined as $H(q, p, t)=K(Q(q, p, t), P(q,...
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1answer
240 views

Time Dependence on Landau & Lifshitz's Proof of Poisson's Bracket Canonical Invariance

I'm reading Landau & Lifshitz's Mechanics and, at a certain point when discussing canonical transformations, they prove that Poisson brackets are canonical invariants. The proof starts with ...
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2answers
283 views

Generating function for canonical transformation

Short version: I've been reading through some notes on integrable systems/Hamiltonian dynamics, and am stuck on a problem: Show that the coordinate transformation derived via the generating function ...
5
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1answer
128 views

Are all time-independent Hamiltonian systems related locally via time-independent canonical transformation?

So recently I've been doing some self-study on canonical transformations and relating together different Hamiltonian systems. I've found this paper (PDF) with a remarkable result showing that any two ...
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1answer
164 views

If $(q,p)$ to $(Q,P)$ is a canonical transformation, then does this imply $(Q,P)$ to $(q,p)$ is also?

If $(q,p)$ to $(Q,P)$ is a canonical transformation, then does this imply $(Q,P)$ to $(q,p)$ is also, assuming Hamilton's equations hold for the coordinates $(q,p)$? This seems like it should be true ...
3
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1answer
197 views

Sufficient conditions for a mapping to be canonical in Hamiltonian Mechanics

My professor mentioned: A simple way of testing whether a mapping $(q,p)$ to $(Q,P)$ is canonical is by examining: $$P · dQ − p · dq$$ and if it equals to $dA$ (a differential) then it is canonical. ...
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0answers
251 views

Showing time-dependent transformations are canonical

In Goldstein's mechanics book he defines a canonical transformation as a transformation from $q, p$ to $Q, P$ with $$ Q = Q(q,p,t) \tag{9.4a} $$ and $$ P = P(q,p,t) \tag{9.4b} $$ such that if $H$ ...
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1answer
169 views

Are all canonical transformations unitary transformations?

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates $(q, p, t) \rightarrow (Q, P, t)$ that preserves the form of Hamilton's equations. Now in quantum mechanics ...
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1answer
116 views

Independent Quantities in Canonical Transformations

I was looking through some lecture slides and I came across this page: I understand that the equation highlighted blue (top right corner) is obtained from the Principle of Least Action. Given a ...
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1answer
102 views

A paradox about canonical transform preserving Poisson bracket?

Let $q,p$ denote the position and momentum. Consider a transform generated by $g$: $q' = q + \epsilon \{q,g\}---(1a)$ $p' = p + \epsilon \{p,g\}---(1b)$ Then: $\{q',p'\} = \{q,p\}+o(\epsilon^2)+\...
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2answers
96 views

How generating function and scale transformation is coming in Hamiltonian?

As I was reading Goldstein, there is the Hamiltonian $H$ such that, $$\delta \int_{t1}^{t2} (p_i\dot q_i - H(q, p ,t)) dt = 0, \tag{9.7}$$ and Kamiltonian $K$, $$\delta \int_{t1}^{t2} (p_i\dot q_i - K(...
3
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0answers
109 views

Canonical transformation of spacetime or under field theory formalism

Is there a standard way to define Canonical Transformations in the case of field theory or spacetime? I saw that under ADM formalism it is possible to define a generating function: $$ G(t)= \sum_i p_i ...
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2answers
101 views

Problem on deriving canonical transformation condition

I'm trying to compute how a canonical transformation should be, given that preserve the symplectic form and trying to recover the condition on the Poisson Bracket. I then start with $$\omega=\stackrel{...
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1answer
54 views

Noetherian versus Hamiltonian symmetries

There are 2 ideas of symmetry I have seen in Classical Mechanics: Noetherian symmetry: Here they discuss infinitesimal point transformations where only position coordinates (and their derivatives ) ...
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1answer
68 views

One parameter-groups and coordinate transformations in phase-space

I have given a function $$G=p_1q_1 - p_2q_2$$ on a 4-dimensional phase-space. This function $G$ commutes with the Hamiltonian $$H= \frac{p_1p_2}{m} + m\omega^2q_1q_2.$$ It generates a flow $$(\vec{q},\...
3
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1answer
80 views

Intuition about non-invariance of the Hamiltonian in canonical transformation

Suppose $q={\{q_i\}}_{i=1}^n$ is the set of generalized coordinates of a dynamical system. $L(q,\dot q,t)$ is the Lagrangian of the system. Now we make coordinate transformation $Q_i=Q_i(q,t)$. Then ...
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1answer
58 views

How is the relationship between the old and the new canonical variables justified?

In Classical Hamiltonian Mechanics, a canonical transformation of the phase-space coordinates $(p,q,t) \to (P,Q,t)$ is such that the general form of Hamilton's equations is followed and Hamilton's ...
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1answer
30 views

Generating Functions for Extended Canonical Transformations

From Goldstein we have that, for non-extended ($\lambda =1 $), the generating function of third type is $$F = F_3(p,Q,t) + q_ip_i.\tag{1}$$ Although I found it hard to see if that would hold true also ...