I am relatively new to Lagrangian and Hamiltonian dynamics. I am aware of how to form the equations of motion using the Legendre Transformation. I, however, have one fundamental question and I was hoping if someone could help me with this.

On performing the Legendre transformation $$H = p^T\dot{q} - L \tag{1}$$ where $p$ is the conjugate momenta, I was under the impression that $p$ and $q$ would be independent variables. However, using $$p = \frac{\partial L}{\partial \dot{q}}\tag{2}$$ gives $p=p(q,\dot{q})$, which implies that $p$ and $q$ are not independent. Is this a non-canonical transformation? Or am I doing something wrong? How do I work around this?


OP's eq. (2) is the definition of Lagrangian momentum, and is not part of a canonical transformation. Canonical transformations are only defined within the Hamiltonian formulation, not in the Lagrangian formulation.

| cite | improve this answer | |
  • $\begingroup$ Thank you. I understand now that this is not a canonical transformation. My main concern was on whether we must require that $p$ and $q$ be independent variables. Which leads me to think that a phase-space analysis in the $p-q$ space may not be as useful in understanding the dynamics of the system in finding out certain properties as much as the $q-\dot{q}$ space. $\endgroup$ – Roshan Thomas Eapen Jun 18 '18 at 14:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.