# Lagrangian of 3 masses connected by springs, non-parallel.

• Daley192303
In summary, when modeling non-linear linkages of three masses and two springs, the Lagrangian can be written using vectors instead of lengths to account for the constant angle of intersection between the springs. However, additional considerations such as Young's modulus may be necessary to accurately model the flexing and bending of the springs.
Daley192303
Writing the Lagrangian for 3 masses and 2 springs in a line is easy.

KE=1/2(m*v^2)

L=KE(m1)+k/2(l1-(x2-x1))^2+KE(m2)+k2/2[L2-(x3-x2)]^2+KE(m3)

However, I wish to model non-linear linkages of the above 3 masses and 2 springs.

Suppose that the second spring (m2-m3) is angle θ away from the axis of the first spring (m1-m2).

I am quite daunted by the flexing or bending of the springs.

Daley192303 said:
Writing the Lagrangian for 3 masses and 2 springs in a line is easy.

KE=1/2(m*v^2)

L=KE(m1)+k/2(l1-(x2-x1))^2+KE(m2)+k2/2[L2-(x3-x2)]^2+KE(m3)

However, I wish to model non-linear linkages of the above 3 masses and 2 springs.

Suppose that the second spring (m2-m3) is angle θ away from the axis of the first spring (m1-m2).

I am quite daunted by the flexing or bending of the springs.
If you simply replace the lengths by vectors, that should work. The angle will not be constant, right?

@haruspex Good question about the angle being constant. The angle of intersection should be constant for a given simulation, however the springs themselves may bend.

Using vectors sounds promising. I am not sure if the vectors (alone) would contain the necessary information given Young's modulus, which is a variation of Hook's Law that applies to the flexing that would occur given the constant angle of intersection, which I should have mentioned to begin with.

The tricky thing about Young's modulus is that while the angles between the springs may be constant at the central mass, the relative angles between the three masses is variable due to flexing of the springs.

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## 1. What is the Lagrangian in this system?

The Lagrangian is a mathematical function that describes the dynamics of a system in terms of its position and velocity. It is defined as the difference between the kinetic energy and potential energy of the system.

## 2. How is the Lagrangian derived for this specific system?

The Lagrangian for a system of 3 masses connected by springs that are not parallel can be derived using the principle of virtual work. This involves considering the forces acting on each mass and their corresponding displacements.

## 3. What are the advantages of using the Lagrangian approach in this system?

The Lagrangian approach allows for a more elegant and concise formulation of the equations of motion compared to traditional methods. It also takes into account the constraints of the system, such as the springs, and allows for a more intuitive understanding of the dynamics.

## 4. Can the Lagrangian be used to study the stability of this system?

Yes, the Lagrangian can be used to study the stability of this system by analyzing the potential energy function. The stability of the system can be determined by examining the nature of the critical points of the potential energy function.

## 5. Are there any limitations to using the Lagrangian in this system?

One limitation is that the Lagrangian approach may not be suitable for systems with complex constraints or non-conservative forces. In such cases, other methods such as the Hamiltonian approach may be more appropriate.

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