Contact interaction in Lagrangians

In summary, the conversation discusses the possibility of integrating out the delta function in a D-dimensional target space for a system involving two point particles coupled by a contact interaction. The issue arises in the variation of the delta function term, where the authors have attempted to write it as a sum over points of intersection between the worldlines of the particles. However, this approach is only valid in D=1 and the correct variation cannot be obtained. The discussion also mentions that there are no space integrals in this problem as it is a gauge-fixed action for point particles in a one-dimensional domain.
  • #1
Illuminatum
8
0
Hi all,

If I take an action involving two point particles coupled together by a delta function contact interaction is it possible to carry out the variation with respect to the fields? For e.g.

[itex]
S = \int dt \frac{1}{2} \dot{x}^{2} + \int \int dt \dot{x}(t) \delta^{D}\left(x(t) - y(t')\right) \dot{y}(t')dt' + \int dt' \frac{1}{2}\dot{y}^{2}

[/itex]

working in, say, D dimensions. I think it will be possible to integrate the delta function out in D=1, but not in higher target space dim. The problem I have is in how to do the variation of the delta function term. Physically it is producing an interaction every time the worldlines of the particles intersect and I've tried writing this as a sum over such points - where [itex]x(t_{0})=y(t')[/itex] - of [itex]\frac{\delta(t - t_{0})}{\dot{x}(t_{0})}[/itex] but this is valid only in D = 1 and I still can't get the variation correct.

Any help would be appreciated.
I
 
Last edited:
Physics news on Phys.org
  • #2
where are your space integral parts.If you are doing a one dimensional problem then you will have D=1,if you are doing a D dimensional one your eqn will be modified and the effect of delta function is to just make x(t)=y(t') in second integral after space integration but nevertheless what you have written for first integral and third (not to mention second one) just does not qualify.
 
  • #3
There are no space integrals - it is a (gauge fixed) action for point particles, proportional to the length of the worldline in D dimensional target space plus some interaction terms. The domain is one dimensional, as for the ordinary relativistic point particle.

Sorry for the confusion
I
 

1. What is contact interaction in Lagrangians?

Contact interaction in Lagrangians refers to the mathematical framework used to describe the interaction between two or more particles in a physical system. It takes into account the forces and energies involved in the interaction and uses the Lagrangian formalism to derive the equations of motion for the system.

2. How is contact interaction different from other types of interactions?

Contact interaction is different from other types of interactions, such as non-contact or long-range interactions, because it is a short-range interaction that only occurs between particles in close proximity. It does not involve the exchange of particles or fields between the particles, unlike non-contact interactions.

3. What are some examples of contact interactions in Lagrangians?

Some examples of contact interactions in Lagrangians include the interaction between two particles connected by a spring, the interaction between two atoms in a molecule, and the interaction between two colliding billiard balls. These interactions are all short-range and do not involve the exchange of particles or fields.

4. How is contact interaction represented in Lagrangian mechanics?

In Lagrangian mechanics, contact interaction is represented by a potential energy term in the Lagrangian function. This potential energy term takes into account the forces involved in the interaction and is used to derive the equations of motion for the system. The Lagrangian function also includes kinetic energy terms for each particle in the system.

5. Why is the concept of contact interaction important in physics?

The concept of contact interaction is important in physics because it allows us to accurately describe and predict the behavior of physical systems. By using the Lagrangian formalism, we can mathematically model the interactions between particles and understand the dynamics of a system. This is crucial for many fields of physics, including mechanics, electromagnetism, and quantum mechanics.

Similar threads

Replies
3
Views
506
  • High Energy, Nuclear, Particle Physics
Replies
3
Views
846
  • High Energy, Nuclear, Particle Physics
Replies
3
Views
1K
  • High Energy, Nuclear, Particle Physics
Replies
6
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
3
Views
2K
Replies
1
Views
759
Replies
19
Views
1K
  • Special and General Relativity
Replies
5
Views
268
  • High Energy, Nuclear, Particle Physics
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
135
Back
Top