Legendre Transformation of Lagrangian density ?

In summary, the conversation discusses the confusion about deriving the Hamiltonian density from the Lagrangian density using the Legendre transformation. It is clarified that the Hamiltonian density is related to the De Donder-Weyl theory and is defined as the canonical stress-energy tensor. Its time component is known as the Hamiltonian and is conserved. The covariant Hamiltonian density has a similar definition but with a different metric.
  • #1
DOTDO
7
0
Hi

I began to study the basics of QED.

Now I am studying Lagrangian and Hamiltonian densities of Dirac Equation.

I'll call them L density and H density for convenience :)Anyway, the derivation of the H density from L density using Legendre transformation confuses me :(

I thought because parameters of them are space-time components, it should be


bandicam 2015-05-03 11-13-28-477.jpg

But I found that this is related to the De Doner - Weyl Theory,

and the H density used in textbook is
bandicam 2015-05-03 11-13-34-326.jpg

where the dot represents time derivative.So, my question is,

why we consider Legendre transformation on only time derivative of phi ?

Is it just 'defined' to consider energy of the system?

Then what does the covariant H density defined in the De Donder - Weyl theory mean?
 
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  • #3
From the lagrangian density [itex]\mathcal{L}[/itex], you can define the canonical stress-energy tensor [itex]\mathcal{T}[/itex] as follows:

[itex]\mathcal{T}^{\mu \nu} = \dfrac{\partial \mathcal{L}}{\partial \partial_\mu \phi_j} \partial^\nu \phi_j - g^{\mu \nu} \mathcal{L}[/itex]

This is a conserved current in the first index:

[itex]\partial_\mu \mathcal{T}^{\mu \nu} = 0[/itex]

Then you can define a hamiltonian density [itex]\mathcal{H}[/itex] in terms of [itex]\mathcal{T}[/itex]:

[itex]\mathcal{H} = \mathcal{T}^{tt} =\dfrac{\partial \mathcal{L}}{\partial \partial_t \phi_j} \partial^t \phi_j - g^{tt} \mathcal{L}[/itex]

This is the same as the expression in the textbook, if you're using the metric where [itex]g^{tt} = +1[/itex] and defining [itex]\dot{\phi_j} = \partial_t \phi_j[/itex]

The hamiltonian is the integral of the hamiltonian density over all space:

[itex]H = \int d^3 x \mathcal{H}[/itex]

It's the hamiltonian, not the hamiltonian density, that is constant:

[itex]\dfrac{d}{dt} H = 0[/itex]
 

Related to Legendre Transformation of Lagrangian density ?

1. What is the Legendre transformation of Lagrangian density?

The Legendre transformation of Lagrangian density is a mathematical operation that transforms a function of one set of variables into a function of a different set of variables. In the context of physics, it is used to transform the Lagrangian density, which is a function of the generalized coordinates and their time derivatives, into the Hamiltonian density, which is a function of the generalized coordinates and their conjugate momenta.

2. Why is the Legendre transformation of Lagrangian density important?

The Legendre transformation of Lagrangian density is important because it allows us to switch from the Lagrangian formalism to the Hamiltonian formalism in classical mechanics. This can simplify the equations of motion and make it easier to solve problems in mechanics. It also has applications in other areas of physics, such as quantum field theory and thermodynamics.

3. How is the Legendre transformation of Lagrangian density performed?

The Legendre transformation of Lagrangian density is performed by taking the partial derivative of the Lagrangian density with respect to the time derivative of each generalized coordinate, and then inverting the resulting equations to solve for the corresponding momenta. These momenta are then substituted back into the Lagrangian density to obtain the Hamiltonian density.

4. What is the difference between the Lagrangian and Hamiltonian formalisms?

The Lagrangian and Hamiltonian formalisms are two different ways of describing the dynamics of a physical system. The Lagrangian formalism is based on the principle of least action, where the equations of motion are derived from a single function, the Lagrangian, which depends on the generalized coordinates and their time derivatives. The Hamiltonian formalism, on the other hand, is based on the Hamiltonian, which is a function of the generalized coordinates and their conjugate momenta. Both formalisms are equivalent and can be used to solve the same problems, but one may be more convenient than the other depending on the specific problem.

5. What are some applications of the Legendre transformation of Lagrangian density?

The Legendre transformation of Lagrangian density has many applications in physics, including classical mechanics, quantum field theory, and thermodynamics. In classical mechanics, it allows us to solve problems using the Hamiltonian formalism, which can be simpler than the Lagrangian formalism in some cases. In quantum field theory, it is used to quantize fields by promoting the classical fields to quantum operators. In thermodynamics, it is used to transform the thermodynamic potentials (e.g. entropy, internal energy) into their corresponding dual variables (e.g. temperature, volume).

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