Linked Questions

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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 ...
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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
173 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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0answers
51 views

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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1answer
510 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
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
59 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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2answers
1k views

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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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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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
598 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
129 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
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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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},\...
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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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