# Poisson Bracket for 1 space dimension field

• Bobdemaths
In summary, the conversation discusses the dynamics of a collection of fields \phi^i (t,x) and their conjugate momenta \pi_i, defined by a Lagrangian involving symmetric and antisymmetric tensors. The momenta are shown to be equal to A_i + b_{ij} \phi ' ^j, where A_i = g_{ij} \dot{\phi}^j. The conversation then moves on to show that the Poisson Bracket \{A_i,A_j\} can be expressed as (\partial_i b_{jk} + \partial_j b_{ki} + \partial_k b_{ij}) \phi ' ^k \delta(x-y), using the canonical relation \{\phi ^i(t
Bobdemaths
Hi,

Suppose you have a collection of fields $\phi^i (t,x)$ depending on time and on 1 space variable, for $i=1,...,N$. Its dynamics is defined by the Lagrangian

$L=\frac{1}{2} g_{ij}(\phi) (\dot{\phi}^i \dot{\phi}^j - \phi ' ^i \phi ' ^j ) + b_{ij}(\phi) \dot{\phi}^i \phi ' ^j$

where $\dot{\phi}^i$ denotes the time derivative of the field ${\phi}^i$ and $\phi ' ^i$ denotes its space derivative, and where $g_{ij}(\phi)$ is a symmetric tensor, and $b_{ij}(\phi)$ an antisymmetric tensor.

One easily computes that the momenta conjugate to the fields $\phi^i (t,x)$ are $\pi_i = A_i + b_{ij} \phi ' ^j$, where $A_i = g_{ij} \dot{\phi}^j$.

Now I would like to show that the (equal time) Poisson Bracket $\{A_i,A_j\}$ is
$\{A_i(t,x),A_j(t,y)\}=(\partial_i b_{jk} + \partial_j b_{ki} + \partial_k b_{ij} ) \phi ' ^k \delta(x-y)$
using the canonical relation $\{\phi ^i(t,x) , \pi_j (t,y)\}=\delta_j^i \delta(x-y)$.

I tried to write $A_i = \pi_i - b_{ij} \phi ' ^j$, and then use $\{\phi ' ^i(t,x) , \pi_j (t,y)\}=\delta_j^i \delta ' (x-y)$. But then I can't get rid of the $\delta '$, and I don't get the $\partial_k b_{ij}$ term.

Am I mistaken somewhere ? Thank you in advance !

OK I found the answer. To get rid of the derivatives of Dirac deltas, I used the identity $f(x) \delta ' (x-y) = f(y) \delta ' (x-y) - f'(x) \delta (x-y)$ which follows from the convolution between delta and $(f \times \varphi)'$ ($\varphi$ is a test function).
This also produces the remaining wanted term.

## What is a Poisson bracket for 1 space dimension field?

A Poisson bracket for 1 space dimension field is a mathematical operation that is used to calculate the time evolution of a system in classical mechanics. It is represented by the symbol {f, g}, where f and g are functions of the system's position and momentum.

## How is a Poisson bracket calculated?

A Poisson bracket is calculated by taking the partial derivative of one function with respect to the position variable and multiplying it by the partial derivative of the other function with respect to the momentum variable, then subtracting the same calculation with the functions reversed. This can be represented as {f, g} = (∂f/∂x)(∂g/∂p) - (∂f/∂p)(∂g/∂x).

## What is the physical significance of a Poisson bracket?

A Poisson bracket represents the rate of change of one variable with respect to another in a system. It is used to describe the dynamics of a system and is a fundamental concept in classical mechanics. It is also closely related to the commutator in quantum mechanics.

## Can a Poisson bracket be used for systems with more than 1 dimension?

Yes, a Poisson bracket can be extended to systems with more than 1 dimension. In such cases, the Poisson bracket is represented by the symbol {f, g} = ∑(∂f/∂xi)(∂g/∂pi) - (∂f/∂pi)(∂g/∂xi), where i represents the dimension of the system.

## What are some applications of the Poisson bracket for 1 space dimension field?

The Poisson bracket has many applications in classical mechanics, such as in Hamiltonian dynamics, symplectic geometry, and canonical transformations. It is also used in quantum mechanics to define the Heisenberg equations of motion and to study the quantum-classical correspondence. Additionally, it has applications in statistical mechanics and fluid dynamics.

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