**Book:** Classical Mechanics 3rd Edition

**Author(s):** Herbert Goldstein, Charles P. Poole, John L. Safko

So, I have tried solving some of the problems of the Chapter 9 of Goldstein Classical mechanics.

You can download the pdf version here: Solutions Goldstein Chapter 9

I have also embedded the pdf below as well as posted them in this blog post.

**CHAPTER 9 – CANONICAL TRANSFORMATIONS**

**DERIVATIONS:**

**9.4. Show directly that the transformation**

**is canonical.**

**9.4. Sol.**

We are given a transformation as follows,

We know that the fundamental **Poisson Brackets **of the transformed variables have the same value when evaluated with respect to any *canonical* coordinate set. In other words **the fundamental Poisson Brackets are invariant under canonical transformation.**

Therefore, in order for the given transformation to be canonical, the Poisson Bracket of * Q,P* with respect to

**q & p**should be equal to 1.

Using the formula for **Poisson Bracket**,

Hence Proved.

**9.5. Show directly that for a system of one degree of freedom, the transformation**

**is canonical, where α is an arbitrary constant of suitable dimensions.**

**9.5. Sol.**

We are given a transformation as follows,

We know that the fundamental **Poisson Brackets **of the transformed variables have the same value when evaluated with respect to any *canonical* coordinate set. In other words **the fundamental Poisson Brackets are invariant under canonical transformation.**

Therefore, in order for the given transformation to be canonical, the Poisson Bracket of * Q,P* with respect to

**q & p**should be equal to 1.

Using the formula for **Poisson Bracket**,

**Hence Proved.**

**9.6. The transformation equations between two sets of coordinates are**

**(a) Show directly from these transformation equations that Q,P are canonical variables if q and p are.
(b) Show that the function that generates this transformation is**

**9.6 Sol. (a)**

We are given a transformation as follows,

We can re-write the second equation as:

We know that the fundamental **Poisson Brackets **of the transformed variables have the same value when evaluated with respect to any *canonical* coordinate set. In other words **the fundamental Poisson Brackets are invariant under canonical transformation.**

Therefore, in order for the given transformation to be canonical, the Poisson Bracket of * Q,P* with respect to

**q & p**should be equal to 1.

Using the formula for **Poisson Bracket**,

**Hence Proved.**

**(b)**

We are given the following generating function of the * F_{3} *type:

For a generating function of

*type,*

**F**_{3}**is given as:**

*q*and * P* is given as:

Plugging the value of Q from eq(i) in the above equation,

Therefore, the given generating does in fact generate the given transformation.

**9.8. Prove directly that the transformation**

**is canonical and find a generating function.**

**9.8. Sol.**

We are given a transformation as follows,

We know that the fundamental **Poisson Brackets **of the transformed variables have the same value when evaluated with respect to any *canonical* coordinate set. In other words **the fundamental Poisson Brackets are invariant under canonical transformation.**

Therefore, in order for the given transformation to be canonical, * Q,P* should satisfy the following condition:

i.e. we need to prove,

and

Using the formula for **Poisson Bracket**,

Now,

**Hence the given transformation is canonical.**

**9.10. Find under what conditions**

**where α and β are constants, represents a canonical transformation for a system of one degree of freedom, and obtain a suitable generating function. Apply the transformation to the solution of the linear harmonic oscillator**.

**9.10.Sol.**

We are given a transformation as follows,

We know that the fundamental **Poisson Brackets **of the transformed variables have the same value when evaluated with respect to any *canonical* coordinate set. In other words **the fundamental Poisson Brackets are invariant under canonical transformation.**

* Q,P* with respect to

**q & p**should be equal to 1.

Using the formula for **Poisson Bracket**,

But for canonical transformation.

which is the condition for the given transformation to be canonical.

**9.11. Determine whether the transformation**

**is canonical.
Sol.9.11**

We are given a transformation as follows,

We know that the fundamental

**Poisson Brackets**of the transformed variables have the same value when evaluated with respect to any

*canonical*coordinate set. In other words

**the fundamental Poisson Brackets are invariant under canonical transformation.**Therefore, in order for the given transformation to be canonical, the Poisson Bracket of * Qi,Pi* with respect to

**q & p**should be equal to 1.

i.e. we need to prove,

Using the formula for **Poisson Bracket**,

Now,

Hence, the given transformation is canonical.

**9.14. Prove that the transformation**

**is canonical, by any method you choose. Find a suitable generating function that will lead to this transformation.
Sol.9.14.**

We are given a transformation as follows,

We know that the fundamental

**Poisson Brackets**of the transformed variables have the same value when evaluated with respect to any

*canonical*coordinate set. In other words

**the fundamental Poisson Brackets are invariant under canonical transformation.**Therefore, in order for the given transformation to be canonical, the Poisson Bracket of * Qi,Pi* with respect to

**q & p**should be equal to 1.

i.e. we need to prove,

Using the formula for **Poisson Bracket**,

Now,

Hence the given transformation is canonical.

**9.15. (a) Using the fundamental Poisson Brackets find the values of α and β for which the equation**

**represents a canonical transformation.
(b) For what values of α and β do these equations represent an extended canonical transformation? Find a generating function of the F _{3} form for the transformation.
(c) On the basis of part (b), can the transformation equations be modified so that they describe a canonical transformation for all values of β?**

**Sol.9.15.(a)**

We are given a transformation as follows,

**Poisson Brackets **of the transformed variables have the same value when evaluated with respect to any *canonical* coordinate set. In other words **the fundamental Poisson Brackets are invariant under canonical transformation.**

* Q,P* with respect to

**q & p**should be equal to 1.

Using the formula for **Poisson Bracket**,

**9.17. Show that the Jacobi identity is satisfied if the Poisson Bracket sign stands for the commutator of two square matrices.**

**Show also that for the same representation of the Poisson bracket that**

**Sol.9.17.**

We are given that the Poisson bracket sign stands for the commutator of two square matrices:

We need to show that the Jacobi identity is satisfied by the new(above) definition of Poisson Brackets.

Therefore, we need to prove:

Proof:

using the new definition of P.B.

Note: We can’t simply cancel ABC by ACB as these are matrices and hence their product isn’t commutative.

Similarly,

and,

Adding (i), (ii) & (iii), we get

Hence, Jacobi Identity is satisfied by the commutator of two square matrices.

Next, we are asked to prove that,

Proof:

from the given definition of P.B.

adding and subtracting

Hence Proved.

**9.21. (a) For a one-dimensional system with the Hamiltonian**

**show that there is a constant of motion**

**(b) As a generalization of part (a), for motion in a plane with the Hamiltonian**

**where p is the vector of the momenta conjugate to the Cartesian coordinates, show that there is a constant of motion**

**(c) The transformation Q=λq, p=λ is obviously canonical. However, the same transformation with with t time dilation, Q=λq, p=λP, t’=λ ^{2}t, is not. Show that, however, the equaitons of motion for q and p for the Hamiltonian in part(a) are invariant under this transformation. The constant of motion D is said to be associated with this invariance.
Sol.9.21. a)**

where we have used,

Therefore, D is a constant of motion.

**9.22. Show that the following transformation is canonical by using Poisson Bracket:**

**Sol.9.22.**

**9.23. Prove that the following transformation from (q,p) to (Q,P) basis is canonical using Poisson Bracket**

**Sol.9.23. NOTE:** There seems to be some typo in the question as the Poisson Bracket of (Q,P) with respect to (q,p) doesn’t come out to be 1.

**9.24. Prove that the transformation defined by Q=1/p and P=qp ^{2} is canonical using Poisson Bracket.
Sol.9.24.**

**9.30.(a) Prove that the Poisson Bracket of two constants os motion is itself a constant of motion even when when the constants depend on time explicitly.
(b) Show that if the Hamiltonian and a quantity F are constants of motion, then the nth partial derivative F with respect to t must also be a constant of motion.
(c) As an illustration of this result, consider the uniform motion of a free particle of mass m. The Hamiltonian is certainly conseerved, and there exists a constant of motion **

Show by direct computation od the Po

**Sol.9.30.**

(a)

(a)

Let the two constants of motion be A(t) & B(t).

Then since A(t) and B(t) are given to be constants of motion, therefore

and

Let’s evaluate the second term in the R.H.S.:

from (i) and (ii) we have

Plugging these back in eq(iv)

Plugging the above back in eq(iii)

Now using Jacobi’s Identity,

(b)

Given:

and,

Therefore, H is not a function of time.

We are also given that,

Now, let’s have a look at the equation of motion of the *n*th partial derivative of F:

the second term on the R.H.S. is clearly zero as H is independent of time as shown earlier already.

Therefore, the nth partial derivative of F is a constant of motion.

(c)

For a free particle of mass m:

Hence Proved.