What is the physical significance of these transformations?

Question

What is the physical significance of coordinate trasformations in phase space that are canonical but do not transform $q$s and $p$s contravariantly w.r.t each other?

For instance, the following families of linear transformations are canonical (their Jacobian matrix is symplectic)

$$\begin{cases} P_1=\frac{p_1-q_1+a(p_2-q_2)}{\sqrt{2}} \\ P_2=\frac{(p_2-q_2)}{\sqrt{2}} \\ Q_1=\frac{(p_1+q_1)}{\sqrt{2}} \\ Q_2=\frac{p_2+q_2-a(p_1+q_1)}{\sqrt{2}} \\ \end{cases} $$

where $a$ is a real parameters.

They map for instance the Hamiltonian $H=\frac{1}{2}(P_1Q_1+P_2Q_2)^2$ into $H=\frac{1}{8}(p_1^2-q_1^2+p_2^2-q_2^2)^2$.

However, if I assume that $H$ is the Legendre transform of a Lagrangian $L(q_1,q_2,\dot{q_1},\dot{q_2})$, being $p_i=\frac{\partial L}{\partial \dot{q_i}}$, I also need $P=(M^t)^{-1}p$ ($M=\frac{\partial(Q_1,Q_2)}{\partial(q_1,q_2)}$ is the Jacobian). This forces the families of transformations above to the single linear transformation where $a=0$.

My questions now are:

• what is the physical significance (if any) of the tranformations with $a\neq 0$?
• why requiring the transformation to be canonical does not yield automatically $a=0$?

I am asking this because any transformation, even with $a\neq 0$, being symplectic, hence canonical, should guarantee that the evolution of a system $\dot x=J \nabla H(p,q)$ transforms into $\dot X=J \nabla H(P,Q)$.

Answer 1

They map for instance the Hamiltonian $H=\frac{1}{2}(P_1Q_1+P_2Q_2)^2$ into $H=\frac{1}{8}(p_1^2-q_1^2+p_2^2-q_2^2)^2$.

Without the square in the Hamiltonian, this would be just a transformation to normal coordinates/normal modes, i.e., decomposing oscillations into a superposition of independent oscillators: $$ H=P_1Q_1+P_2Q_2\longrightarrow H=\frac{1}{2}(p_1^2-q_1^2+p_2^2-q_2^2) $$ As a non-quadratic Hamiltonian generally describes non-linear system, it is unlikely that there is some general meaning to it - the transformation is likely inspired by that of a quadratic Hamiltonian, in order to simplify the analysis.

Note that transformation to normal modes is closely related to those for classifying second-order curves/conic sections, so perhaps the meaning can be inferred from a similar transformation for fourth-order curves.

Finally, not that physical meaning/significance is a rather intuitive concept, related to a specific problem, the objective of study, etc. - it is not hard-coded in mathematical form used to describe the problem (which can be the same for very different system - mechanical, electromagnetic, atomic, etc.)

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