Canonical Transformations

  • Gennady StupakovEmail author
  • Gregory Penn
Part of the Graduate Texts in Physics book series (GTP)


One of the benefits of the Lagrangian approach to mechanical systems is that we can choose the generalized coordinates as we please. We have seen that once we select a set of coordinates \(q_i\) we can define the generalized momenta \(p_i\) according to Eq. ( 1.26) and form a Hamiltonian ( 1.28). We could also have chosen a different set of generalized coordinates \(Q_i=Q_i(q_k, t)\), expressed the Lagrangian as a function of \(Q_i\), used Eqs. ( 1.26) and ( 1.28), and obtained a different set of momenta \(P_i\) and a different Hamiltonian \(H'(Q_i,P_i, t)\). Although mathematically different, these two representations are physically equivalent — they describe the same dynamics of our physical system. Understanding the freedom that we have in the choice of the conjugate variables for a Hamiltonian is important: a judicious choice of the variables could allow us to simplify the description of the system dynamics.

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.SLAC National Accelerator LaboratoryStanford UniversityMenlo ParkUSA
  2. 2.Lawrence Berkeley National LaboratoryBerkeleyUSA

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