Linked Questions

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0answers
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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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0answers
41 views

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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4answers
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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 ...
10
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3answers
1k 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 ...
10
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4answers
1k 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 ...
9
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2answers
2k 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
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1answer
180 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. ${\...
5
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2answers
719 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 \...
3
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2answers
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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 ...
5
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1answer
708 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(...
2
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1answer
1k 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 ...
5
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1answer
472 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).
3
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1answer
829 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 ...
4
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2answers
731 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 ...
6
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2answers
385 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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