# Classical Mechanics: Canonical transformation problem

• genius2687
In summary, the conversation discusses transforming equations to determine if they are canonical. The equations provided have no time dependence and can be considered canonical if certain conditions are met. The attempted solution involves finding the partial derivatives for each equation and solving for the corresponding variables, but an error is made in assuming that the derivatives are equal to their reciprocals.
genius2687

## Homework Statement

Show directly that the transformation; Q=log(1/q*sinp), P=q*cotp is canonical.

## Homework Equations

Since these equations have no time dependence, the equations are canonical if (with d denoting a partial derivative)

dQ_i/dq_j = dp_j/dP_i, and dQ_i/dp_j = -dq_j/dP_i

## The Attempt at a Solution

With

Q=log(1/q*sinp), dQ/dq = -1/q

P=q*cotp => p=tan^-1(q/P), dp/dP = -q/(p^2+q^2).

The first problem I encounter is that -1/q not= -q/(p^2+q^2).

With dQ/dp = cotp, and -dq/dP = -1/(dP/dq) = -cotp

so, cotp not= -cotp.

Last edited:
It worked for me. I solved for

cosp = Pe^Q

q = sinp/e^Q = sqrt[1 - (Pe^Q)²]/e^Q

and took the derivatives.

I know what I did now. For partial derivatives, dx/dy not= 1/(dy/dx). I falsely made that assumption.

## 1. What is a canonical transformation?

A canonical transformation is a mathematical transformation that preserves the form of Hamilton's equations in classical mechanics. It transforms one set of canonical variables to another set that describes the same physical system.

## 2. Why are canonical transformations important?

Canonical transformations are important because they provide a way to simplify the equations of motion in classical mechanics. They also reveal symmetries in a system and can be used to find conserved quantities, such as energy and momentum.

## 3. How do you determine if a canonical transformation is valid?

A canonical transformation is valid if it satisfies the canonical transformation equations, which relate the old and new sets of canonical coordinates and momenta. These equations must also be reversible, meaning that the inverse transformation must also be valid.

## 4. Can a canonical transformation change the physical properties of a system?

No, a canonical transformation does not change the physical properties of a system. It only changes the way the system is described mathematically by transforming the canonical variables and momenta.

## 5. What is the difference between a canonical transformation and a Legendre transformation?

A canonical transformation is a type of transformation that preserves the form of Hamilton's equations, while a Legendre transformation is a specific type of canonical transformation that transforms between the Lagrangian and Hamiltonian formulations of classical mechanics. In other words, a Legendre transformation is a special case of a canonical transformation.

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