Canonical and conjugate momentum

[igraviton]

what is the difference between canonical and conjugate momentum.. ? what is its physical significant.. I was reading classical mechanics by Goldstein but could understood this terms

 

[Bill_K]

Exactly the same thing. You might use the term conjugate momentum when you're referring to the canonical momentum which is conjugate to a particular coordinate.

 

[igraviton]

Hi Bill_k,

Thanks for reply but what that means ? "conjugate to a particular coordinate"... physical intrepretation

 

[tom.stoer]

The canonical conjugate momentum p is derived via a derivative of the action w.r.t. the time derivative of a generalized coordinate q. Then {q,p}=1. The physical interpretation of p depends on the interpretation of q.

 

[PJC01]

Canonical momentum and conjugate momentum are two important concepts in classical mechanics. They are both related to the motion of a particle or system of particles, but they have different definitions and physical significance.

Canonical momentum is defined as the momentum of a particle or system of particles in terms of generalized coordinates and their derivatives. It is denoted by p and is given by the expression p = ∂L/∂q̇, where L is the Lagrangian of the system and q̇ is the time derivative of the generalized coordinate q. Canonical momentum is a fundamental quantity in Hamiltonian mechanics, which is a reformulation of classical mechanics that uses the Hamiltonian function to describe the dynamics of a system.

On the other hand, conjugate momentum is defined as the momentum conjugate to a particular coordinate in a system. It is denoted by π and is given by the expression π = ∂L/∂q̇, where L is the Lagrangian of the system and q̇ is the time derivative of the coordinate q. Conjugate momentum is used in the Hamiltonian formalism to derive the equations of motion for a system.

The main difference between canonical and conjugate momentum is that canonical momentum is a general concept that can be applied to any system, while conjugate momentum is specific to a particular coordinate in a system. Additionally, while canonical momentum is based on the Lagrangian of a system, conjugate momentum is based on the Hamiltonian of a system.

The physical significance of these two concepts lies in their role in describing the dynamics of a system. In classical mechanics, momentum is a fundamental quantity that represents the motion and inertia of a particle or system. By using canonical and conjugate momentum, we can derive the equations of motion for a system and study its behavior.

In summary, canonical momentum and conjugate momentum are important concepts in classical mechanics that are used to describe the motion of a particle or system. While they have different definitions and roles, they both play a crucial role in understanding the dynamics of a system.