Masses and springs in R^3XR^2 and K.G. Eq. on a lattice?

In summary, there is a 5 dimensional lattice in R^5 = R^3XR^2 where all lattice points have a point mass and are connected by springs in all 5 dimensions to their nearest neighbors. The motion of the mass points is limited to a 2 dimensional subspace, R^2, and the springs in R^2 are stiffer than the remaining springs. The goal is to create a 3 dimensional mass spring lattice system where the masses move only in a 2 dimensional tangent space at each point, similar to a 2 dimensional harmonic oscillator coupled with springs to nearest neighbors in 3 dimensions. The question is whether the lagrangian for this system is similar to the lagrangian for the Klein Gordon
  • #1
Spinnor
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Let us have a 5 dimensional lattice in R^5 = R^3XR^2, where at each lattice point we have a point mass and all mass points are linked with springs in all 5 dimensions (edit, to nearest neighbors). Require that motion of the mass points is restricted to a two dimensional subspace, R^2, of R^5. Further let the springs in R^2 be much stiffer then the remaining springs. Make further assumptions as required so the problem comes out right.

Edit, something does not smell right, should of thought more, think I am close but I think the above is wrong, damn.

Edit, we want only one mass point in the subspace R^2?

4th edit, we want a 3 dimensional mass spring lattice system with movement of the masses only in some 2 dimensional tangent space at each mass point and that movement is that of 2 dimensional harmonic osscilator coupled with springs to nearest neighbors in 3d? Put a 2d H.O. at each lattice point in 3d and couple the movement to nearest neighbors in 3d with springs? 5th edit, all done in a 5d space?

Is the lagrangian of this system similar to the lagrangian for the Klein Gordon equation on a lattice?

Is there a electrical circuit equavalent using capacitors and inductors of the above system?

If true can one expand on the ideas above and come up with a mass/spring system on some space whose lagrangian is the same as the lagrangian that gives the Dirac eq. on a lattice?

Let us ignor relativity (set c to infinity?).

Thanks for any help!
 
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  • #2
A sketch of the system above may be less confusing then my ramblings above.

K%2BG%2Beq%2B002.jpg
 

1. What is a mass-spring system?

A mass-spring system is a physical system that consists of a mass or multiple masses connected by springs. The motion of the system is governed by the interactions between the masses and the springs, and can be described using mathematical equations.

2. How is a mass-spring system represented in R^3XR^2?

In R^3XR^2, a mass-spring system is represented as a set of points in 3-dimensional space (R^3) connected by lines in 2-dimensional space (R^2), which represent the springs. This representation allows for visualization and analysis of the system's motion in both 3D and 2D space.

3. What is the significance of the K.G. Eq. on a lattice in mass-spring systems?

The K.G. Eq. (Kinetic Growth Equation) on a lattice is a mathematical equation that describes the motion and growth of a mass-spring system on a lattice, which is a discrete grid-like structure. This equation is used to model and analyze the behavior of complex mass-spring systems in various fields of science and engineering.

4. What is the role of the spring constant in mass-spring systems?

The spring constant is a measure of the stiffness of a spring and determines the amount of force required to stretch or compress the spring. In a mass-spring system, the spring constant affects the strength and frequency of the oscillations of the masses, and can be adjusted to alter the behavior of the system.

5. How are mass-spring systems used in real-world applications?

Mass-spring systems have many practical applications, such as in mechanical engineering, where they are used in the design of suspension systems and shock absorbers. They are also used in physics and chemistry to model the behavior of atomic and molecular structures. In addition, mass-spring systems are used in computer animation and video game programming to simulate realistic movements and interactions.

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