Definition of a canonical variable

[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.

 

[smallphi]

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':

$$\frac{dq}{dt} = \frac{\partial H}{\partial p}$$

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

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:

$$\frac{dQ}{dT} = \frac{\partial K}{\partial P}$$

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

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

 

[lightarrow]

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

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

$$p_i = \frac{\partial L}{\partial \dot {q_i}}$$

The set of all {$$q_i$$} and {$$p_i$$} are the canonical variables. The Hamiltonian function is defined as:

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

Then canonical equations come from that.

 

[gunnpark]

actually definition above is not completely true. i believe that definition is given in goldstein's book.
but in that logic, you can not distinguish one dynamical system from another. (Both have canonical looking equation) So if you write down two equations and start claiming both of them are for one dynamical system, then with your logic you there is no way to refute them. for example, harmonic oscillator has canonical variable q,p, and Hamiltonian H. say, gravitational system has P,Q, K. Then obviously P,Q,K are not canonical variables and Hamiltonian for harmonic oscillator. But according to above definition, they are.

Additional element to complete the definition is that you should have generator F, that connects two sets coherently. So you should think of canonical variables as a member of family of variables with Hamiltonian satisfying Hamiton's equation AND connected with each other through fenerator F.