Linked Questions

15
votes
4answers
3k views

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 ...
10
votes
3answers
974 views

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 ...
9
votes
4answers
480 views

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 ...
7
votes
2answers
945 views

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 ...
5
votes
1answer
633 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(...
3
votes
2answers
462 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 \...
1
vote
1answer
702 views

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 ...
4
votes
1answer
309 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).
2
votes
1answer
455 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
votes
2answers
423 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 ...
5
votes
1answer
328 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 ...
2
votes
3answers
135 views

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 ...
2
votes
1answer
427 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 ...
4
votes
1answer
242 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 ...
3
votes
2answers
185 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 ...

15 30 50 per page