Discussion Overview
The discussion revolves around the relationship between canonical transformations and Hamiltonians, specifically in the context of a harmonic oscillator. Participants explore the implications of these transformations on the forms of the Hamiltonians involved and the conditions under which they can be considered equivalent.
Discussion Character
- Conceptual clarification
- Debate/contested
Main Points Raised
- One participant questions how K(P, Q) can equal H(p, q) when they appear in different forms, suggesting a need for clarification on their equivalence.
- Another participant notes that K(P, Q) and H(p, q) have different forms due to the use of different parameters, indicating that one set uses Cartesian coordinates while the other uses polar coordinates.
- A participant seeks to clarify that K(P, Q) = H(p, q) means evaluating H at specific values of p and q should yield the same numerical result as evaluating K at the transformed values of P and Q, without implying that K and H share the same functional form.
- It is mentioned that if K and H had the same functional form, it would imply a symmetry in the canonical transformation.
- Another participant points out that Q does not appear in Hamiltonian K, labeling it as a cyclic coordinate, and provides the Hamilton equation of motion for the conjugate momentum.
Areas of Agreement / Disagreement
Participants express uncertainty regarding the equivalence of K and H, with some agreeing on the numerical equivalence under transformation but disagreeing on the implications for functional forms. The discussion remains unresolved regarding the broader implications of these transformations.
Contextual Notes
Participants highlight that the differences in functional forms may be due to the choice of coordinates and the nature of the canonical transformation, but the implications of these differences are not fully resolved.