Canonical Transformations

In summary, the conversation discusses the relationship between invertible transformations (q,p)↔(Q,P) and Hamiltonian systems. It is shown that for every Hamiltonian H(q,p,t), there exists a K that satisfies the conditions Q˙i=∂K∂PiP˙i=−∂K∂Qi. This is because K should not depend on the derivatives of Q and q. The conversation also discusses the derivation of the "generator" for other pairs of independent phase-space variables using Legendre transformations.
  • #1
MisterX
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Background

For which of the invertible transformations [itex](\mathbf{q}, \mathbf{p}) \leftrightarrow(\mathbf{Q}, \mathbf{P})[/itex]
[itex]\mathbf{Q}(\mathbf {q}, \mathbf {p}, t)[/itex]
[itex]\mathbf{P}(\mathbf{q}, \mathbf {p}, t)[/itex]
is it so that for every Hamiltonian [itex]\mathcal{H}(\mathbf {q}, \mathbf {p}, t)[/itex] there is a [itex]\mathcal{K}[/itex] such that
[tex]\dot{Q}_i = \frac{\partial\mathcal{K}}{\partial P_i} \;\;\;\;\;\;\;\; \dot{P}_i = -\frac{\partial\mathcal{K}}{\partial Q_i}\; ?[/tex]
Stationary action should correspond, and that condition is met if

[itex]\sum p_i\dot{q}_i - \mathcal{H} = \sum P_i\dot{Q}_i - \mathcal{K} + \frac{dF}{dt}[/itex],

since integrating [itex]\frac{dF}{dt}[/itex] results in something only dependent of the endpoints.

Question
Consider this part of Goldstein's Classical Mechanics.

m3icfYL.png


rearranging 9.13 to make this clear:

[tex]\mathcal{K} = \mathcal{H} + \frac{\partial F_1}{\partial t} + \sum_i \dot{Q}_i\left(P_i - \frac{\partial F_1}{\partial Q_i} \right) + \sum_i \dot{q}_i\left(\frac{\partial F_1}{\partial q_i} - p_i\right)[/tex]
I guess I might like this explained a more. Why aren't we able to to have the coefficients of [itex]\dot{q}_i[/itex] or [itex]\dot{Q}_i[/itex] be non-zero, and have the difference absorbed into [itex]\mathcal{K}[/itex] ?
 
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  • #2
it's because K should not depend on the derivates of Q and q
 
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  • #3
I would argue as follows. The Hamiltonian version of the action is
[tex]S[x,p]=\int_{t_1}^{t_2} \mathrm{d} t [\dot{q}^k p_k - H(t,q,p)].[/tex]
The trajectory in phase space is determined as the stationary point of this functional with the boundary values [itex]q(t_1)[/itex] and [itex]q(t_2)[/itex] fixed.

Now if you want to determine new phase-space coordinates [itex](Q,P)[/itex] that describe the same system in the new coordinates by the variational principle, i.e., such that the phase-space trajectories are described by the Hamilton canonical equations, you must have
[tex]\mathrm{d} q^k p_k - \mathrm{d} Q^k P_k - \mathrm{d} t (H-K)=\mathrm{d} f.[/tex]
From this it is clear that the "natural" independent variables for [itex]f[/itex] are [itex]q[/itex], [itex]Q[/itex], and [itex]t[/itex]. Then comparing the differential on each side leads to
[tex]p_k=\frac{\partial f}{\partial q^k}, \quad P_k=-\frac{\partial f}{\partial Q^k}, \quad K=H+\frac{\partial f}{\partial t}.[/tex]
Then you can go over to other pairs of old and new independent phase-space variables in the "generator" using appropriate Legendre transformations.
 
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  • #4
Jul 7, 2013 #1
MisterX
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585
Background

For which of the invertible transformations (q,p)↔(Q,P)
Q(q,p,t)
P(q,p,t)
is it so that for every Hamiltonian H(q,p,t) there is a K such that
Q˙i=∂K∂PiP˙i=−∂K∂Qi?Is It show that K is dependent upon (P, Q) not in Derivative of (P, Q)? please.

also show that
Q˙i=∂K∂PiP˙i=−∂K∂Qi?
 
  • #5


I can explain this concept as follows:

In classical mechanics, canonical transformations are a mathematical tool used to transform the coordinates and momenta of a system while preserving the Hamiltonian equations of motion. In other words, they allow us to change the variables used to describe a system without changing its physical behavior.

In the given equation, we see that for a canonical transformation to exist, there must be a function \mathcal{K} that satisfies the conditions for both \mathbf{Q} and \mathbf{P}. This function is known as the generating function of the transformation.

Now, in order for the stationary action to correspond to the Hamiltonian, we need to have a specific form for \mathcal{K}. This is where the rearranged equation 9.13 comes in. We can see that \mathcal{K} is equal to the Hamiltonian plus some additional terms. These terms are dependent on the transformation and are necessary for the stationary action to be satisfied.

In particular, we see that the coefficients of \dot{q}_i and \dot{Q}_i cannot be non-zero because then the stationary action would not correspond to the Hamiltonian. This is because the Hamiltonian equations of motion require the terms involving \dot{q}_i and \dot{Q}_i to be equal to each other. If the coefficients were different, then the stationary action would not satisfy this condition and therefore would not correspond to the Hamiltonian.

In summary, the equation 9.13 shows us that for a canonical transformation to exist, there must be a specific form for \mathcal{K} that includes additional terms in order for the stationary action to correspond to the Hamiltonian. This is why the coefficients of \dot{q}_i and \dot{Q}_i cannot be non-zero and must be absorbed into \mathcal{K}.
 

1. What is a canonical transformation?

A canonical transformation is a mathematical transformation that preserves the fundamental equations of motion in a physical system. It is a change of variables that transforms the system's original coordinates and momenta into new ones, while maintaining the Hamiltonian equations of motion.

2. How is a canonical transformation different from a general transformation?

A canonical transformation is a special type of transformation that preserves the symplectic structure of the system, which is necessary for maintaining the fundamental equations of motion. In contrast, a general transformation does not necessarily preserve this structure and may result in different equations of motion.

3. What are the applications of canonical transformations?

Canonical transformations are commonly used in classical mechanics and quantum mechanics to simplify the equations of motion and to find new conserved quantities. They are also used in the study of Hamiltonian systems and in the development of new mathematical methods for solving physical problems.

4. How are canonical transformations related to the principle of least action?

Canonical transformations are closely related to the principle of least action, which states that the true path of a physical system is the one that minimizes the action (a quantity related to the system's energy). Canonical transformations preserve this principle, meaning that the equations of motion derived from the transformed coordinates also satisfy the principle of least action.

5. Can canonical transformations be applied to any physical system?

Yes, canonical transformations can be applied to any physical system that can be described using Hamiltonian mechanics. This includes a wide range of classical and quantum systems, such as particles, fields, and rigid bodies. However, the specific form of the transformation will depend on the particular system and its symplectic structure.

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