Poisson Bracket - Constrained system

In summary, the conversation discusses the use of the Dirac Hamiltonian method to drive constraints for a Lagrangian density. The main issue is with calculating a specific type of Poisson Bracket involving the conjugate momentum and its derivative. The response suggests using partial integration and notes the presence of a minus sign in certain cases. The conversation ends with a question about the possible appearance of a minus sign in the second line of the calculation.
  • #1
vnikoofard
12
0
Hi friends

I am trying to drive constraints of a Lagrangian density by Dirac Hamiltonian method. But I encountered a problem with calculating one type of Poisson Bracket:
{[itex]\varphi,\partial_x\pi[/itex]}=?
where [itex]\pi[/itex] is conjugate momentum of [itex]\varphi[/itex]. I do not know for this type Poisson Bracket I can use part-by-part integration or not. I mean
{[itex]\varphi,\varphi\partial_x\pi[/itex]}= -[itex]\varphi[/itex]

Cheeeers!
Vahid
 
Physics news on Phys.org
  • #2
[tex]\{\varphi(x),\pi(y)\} = \delta(x-y)[/tex]
[tex]\{\varphi(x),\partial_y\pi(y)\} = \partial_y\{\varphi(x),\pi(y)\} = \partial_y \delta(x-y)[/tex]
 
  • #3
Thanks very much for response.
I wonder myself maybe appear a minus sign in the second line. Are you sure? Maybe I am confusing this situation with part by part integration!
 
Last edited:
  • #4
The minus sign appears in partial integration or when the derivative is acting on x instead of y:

[tex]\partial_y \delta(x-y) = \delta(x-y)\partial_y[/tex]
[tex]\partial_y \delta(x-y) = -\partial_x \delta(x-y) = -\delta^\prime(x-y)[/tex]
 

What is a Poisson Bracket?

A Poisson Bracket is a mathematical tool used to describe the dynamics of a constrained system. It is a way to calculate how the variables of a system change over time, taking into account the constraints imposed on the system.

How is a Poisson Bracket calculated?

A Poisson Bracket is calculated by taking the partial derivatives of the variables in the system with respect to time and multiplying them by the partial derivatives of the constraints with respect to the variables. This is then subtracted from the partial derivative of the Hamiltonian (a function that describes the energy of the system) with respect to the variables.

What is a constrained system?

A constrained system is a physical system in which the motion or behavior of the system is limited or restricted by certain constraints. These constraints can be imposed by external forces, laws of physics, or the structure of the system itself.

What is the importance of the Poisson Bracket in physics?

The Poisson Bracket is important in physics because it allows us to analyze the dynamics of a constrained system and make predictions about its behavior over time. It is a fundamental tool in the study of classical mechanics and is used in various fields such as celestial mechanics, fluid dynamics, and electromagnetism.

Can the Poisson Bracket be applied to quantum systems?

While the Poisson Bracket was originally developed for classical systems, it has also been extended to quantum mechanics in the form of the Poisson Bracket algebra. However, the interpretation and application of the Poisson Bracket in quantum mechanics differs from its use in classical mechanics.

Similar threads

  • Electromagnetism
Replies
2
Views
803
  • Advanced Physics Homework Help
Replies
6
Views
1K
Replies
20
Views
6K
Replies
4
Views
1K
  • Quantum Physics
Replies
19
Views
3K
  • Classical Physics
Replies
6
Views
5K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Special and General Relativity
Replies
2
Views
901
  • Introductory Physics Homework Help
Replies
6
Views
944
  • Advanced Physics Homework Help
Replies
1
Views
1K
Back
Top