Time dependent canonical transformation

In summary, the question discusses a canonical transformation involving variables P, Q, p, and q. It is stated that the transformation is canonical when {Q,P}=1. However, if a new variable t' is introduced as t' = λ^2t, the transformation is no longer canonical. This raises the question of when {Q,P}=1 can be used to check for canonical transformations, as it appears to still hold true for the second transformation. However, it is noted that the scales of coordinate and momentum must also be taken into account for compatibility.
  • #1
LCSphysicist
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Homework Statement
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Relevant Equations
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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 confused, since the equations of motion remain the same.

So two question:

Why the second transformation is not canonical? And,
When can we use ##\{Q,P\}=1## to check if it is canonical? SInce in the second transformation we still have the same Poisson bracket, but it is not canonical anymore, i am afraid i have been using it unconsciously many times and by coincidence being right.
 
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  • #2
Herculi said:
Homework Statement:: .
Relevant Equations:: .

But, if we add t′=λ2t, the question says that the transformation is not canonical anymore.
[tex]t' = \lambda^2 t [/tex] seems t=1 second corresponds to t'=##\lambda^2## second say one new second. Should p=1 kg m/s correspond to p'=##\lambda^2## kg m / new second ?
 
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  • #3
anuttarasammyak said:
[tex]t' = \lambda^2 t [/tex] seems t=1 second corresponds to t'=##\lambda^2## second say one new second. Should p=1 kg m/s correspond to p'=##\lambda^2## kg m / new second ?
I am afraid i didn't get what you mean.
 
  • #4
Mutual changes in scale of coordinate and momentum, i.e.
[tex]P=\frac{p}{\lambda},Q=\lambda q[/tex]
keep {P,Q}=1 but I am afraid
[tex]P=\frac{p}{\lambda}, Q=\lambda q, T(=t')=\lambda^2 t[/tex]
are not compatible except ##\lambda = \pm 1##.
 
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Related to Time dependent canonical transformation

1. What is a time dependent canonical transformation?

A time dependent canonical transformation is a mathematical tool used in classical mechanics to transform the coordinates and momenta of a system from one set of variables to another set of variables. It preserves the Hamiltonian of the system and preserves the equations of motion. It is useful for simplifying the equations of motion or for finding new conserved quantities in a system.

2. How is a time dependent canonical transformation different from a time independent canonical transformation?

A time dependent canonical transformation involves a transformation that explicitly depends on time, while a time independent canonical transformation does not. This means that the new coordinates and momenta in a time dependent canonical transformation are explicitly functions of time, while in a time independent transformation, they are not.

3. What are some examples of time dependent canonical transformations?

One example of a time dependent canonical transformation is the transformation from Cartesian coordinates to polar coordinates. Another example is the transformation from the standard coordinates and momenta in a harmonic oscillator system to action-angle variables.

4. Can a time dependent canonical transformation be used to change the Hamiltonian of a system?

No, a time dependent canonical transformation does not change the Hamiltonian of a system. It only changes the coordinates and momenta in which the Hamiltonian is expressed. The new Hamiltonian in the transformed coordinates will be equivalent to the original Hamiltonian.

5. How is a time dependent canonical transformation related to the principle of least action?

In classical mechanics, the principle of least action states that the true path of a system is the one that minimizes the action, which is a function of the coordinates and momenta. A time dependent canonical transformation can be used to find new coordinates and momenta that make the action stationary, simplifying the equations of motion and making the principle of least action easier to apply.

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