Lagrangian for fields AND particles?

In summary, the conversation discusses the construction of a Lagrangian for a system of interacting fields and particles. The speaker is interested in how to mix particles and fields in Lagrangian mechanics and asks for an example of a Lagrangian that governs both fields and discrete sources. The conversation also mentions the full action for both electromagnetic fields and dynamic sources and asks for guidance on how to derive the equations of motion from this action using the Euler-Lagrange equations. The speaker requests references to helpful source material on this topic.
  • #1
pellman
684
5
In general what does a Lagrangian for a system consisting of interacting fields and particles look like?

It can't be, for example,

[tex]L=\sum{\frac{1}{2}mv_j^2+A(x_j)\inner v_j}[/tex]

That would be for a system of particles in a fixed, i.e. "background", field. I'm interested in how we can mix particles and fields in Lagrangian mechanics. I know how to write down, as above, the Lagrangian for particles influenced by a field. And I know how to write down a Lagrangian (density) for a field with fixed (continuum) sources. But what does a Lagrangian (density?) that governs both fields and discrete sources look like?

No need to lay out the most general case. Just a simple example will suffice.
 
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  • #2
Ok. No replies. I can take this now to the next step myself. Then maybe someone else can help from there.

Supposedly, the full action for both EM fields and (dynamic) sources is

[tex] -m \int d\tau \sqrt{- g_{\mu \nu} \dot{x}^\mu \dot{x}^\nu} + q \int dx'^4 \int d\tau ~ \delta^4(x'-x(\tau)) \frac{dx^\mu (\tau)}{d\tau} A_\mu - \frac{1}{4} \int d^4 x F^{\alpha \beta} F_{\alpha \beta} [/tex]

See this thread: https://www.physicsforums.com/showthread.php?t=222066

Ok. Now - how do we get the equations of motion from this action? How do we apply the Euler-Lagrange equations to an action of mixed particles and fields?

References to helpful source material would be much appreciated.
 
  • #3


A Lagrangian for a system consisting of both fields and particles is known as a field-particle Lagrangian. It takes into account the dynamics of both the fields and the particles, as well as their interactions with each other.

In general, the Lagrangian for such a system would include terms for the kinetic energy of the particles, potential energy for the particles interacting with the fields, and the Lagrangian density for the fields themselves. It would also include terms for the interactions between the fields and the particles.

One example of a field-particle Lagrangian is the Standard Model of particle physics, which describes the interactions between particles and the electromagnetic, weak, and strong nuclear forces. The Lagrangian for this system includes terms for the kinetic and potential energy of the particles, as well as terms for the gauge fields that mediate the interactions between the particles.

Another example is the Lagrangian for a system of charged particles interacting with an electromagnetic field. This would include terms for the kinetic energy of the particles, the potential energy due to the particles' interactions with the electromagnetic field, and the Lagrangian density for the electromagnetic field.

In general, the Lagrangian for a system of fields and particles will depend on the specific interactions and dynamics of the system. It can be written as a sum of terms for each individual component, with additional terms for their interactions. The resulting equations of motion can then be derived using the principle of least action, just like in traditional Lagrangian mechanics.
 

What is a Lagrangian for fields and particles?

A Lagrangian for fields and particles is a mathematical function that describes the dynamics of a system of particles and fields. It takes into account the interactions and forces between the particles and fields, and allows for the prediction of their behavior over time.

How is a Lagrangian for fields and particles used in physics?

A Lagrangian for fields and particles is used in physics to formulate the equations of motion for a system of particles and fields. It is a powerful tool for predicting the behavior of physical systems and has applications in various fields, including classical mechanics, quantum mechanics, and relativity.

What is the difference between a Lagrangian for fields and particles and a Lagrangian for particles only?

A Lagrangian for fields and particles takes into account the interactions and forces between particles and fields, while a Lagrangian for particles only does not. A Lagrangian for particles only is typically used for systems where the interactions between particles are negligible, while a Lagrangian for fields and particles is necessary for systems where the interactions between particles and fields are significant.

How is a Lagrangian for fields and particles derived?

A Lagrangian for fields and particles is typically derived using the principles of Lagrangian mechanics, which involves finding a function that minimizes the action of the system. This function is then used to derive the equations of motion for the system.

What are some real-world applications of a Lagrangian for fields and particles?

A Lagrangian for fields and particles has numerous applications in physics, including predicting the behavior of particles in accelerators and in the study of fluid dynamics. It is also used in the development of theories and models in quantum field theory, cosmology, and particle physics.

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