Canonical variables

**ehrenfest** · Jun21-07, 03:01 AM

Can someone give me a good definition of a canonical variable? I have seen it in the context of Lagrangians and Hamiltonians. I currently understand it as a "generalization" or an "abstraction" of a regular variable, but there has got to be a better definition.

**smallphi** · Jun22-07, 01:11 AM

First you define the system with a Lagrangian (function of generalized coordinates, their time derivatives and time). From that Lagrangian you form the Hamiltonian (function of generalized coordinates, momenta and time). Those generalized coordinates and momenta are canonical cause the Hamilton equations of motion look in the 'canonical way':

$LaTeX Code: \\frac{dq}{dt} = \\frac{\\partial H}{\\partial p}$

$LaTeX Code: \\frac{dp}{dt} = - \\frac{\\partial H}{\\partial q}$

Now later you may decide to change the generalized coordinates, momenta and even time to other coordinates: (q, p, t) -> (Q, P, T). A change of the variables in general changes the form of the differential equations of motion. The new variables are called canonical if then new equations of motion have the same 'canonical' form albeit with different effective Hamiltonian K:

$LaTeX Code: \\frac{dQ}{dT} = \\frac{\\partial K}{\\partial P}$

$LaTeX Code: \\frac{dP}{dT} = - \\frac{\\partial K}{\\partial Q}$

You can test if a coordinate transformation of (q, p, t) will be canonical by using Poisson brackets.

**lightarrow** · Jun22-07, 10:21 AM

Originally Posted by ehrenfest

Can someone give me a good definition of a canonical variable? I have seen it in the context of Lagrangians and Hamiltonians. I currently understand it as a "generalization" or an "abstraction" of a regular variable, but there has got to be a better definition.

First you have to find the generalized coordinates {

} of the n degrees of freedom system as n parameters which identify univocally the system's state.

Once defined the n generalized coordinates {

} i = 1,..n and the relative lagrangian $LaTeX Code: L(q_i,\\dot {q_i},t)$ , then coniugated momentums are defined as

$LaTeX Code: p_i = \\frac{\\partial L}{\\partial \\dot {q_i}}$

The set of all {

} and {

} are the canonical variables. The Hamiltonian function is defined as:

$LaTeX Code: H(q_i,p_i,t) = \\sum p_i {\\dot {q_i}} - L$

Then canonical equations come from that.