# Degenerating force in Lagrangian mechanics

• laird HT
In summary, The conversation discusses the derivation of equations of motion for three coupled pendula, where the central pendulum has a resistive degenerative force due to submersion in a liquid. The individual has already calculated the normal modes without the resistance, using the lagrangian approach. They are seeking help with integrating the resistance into the system, and it is suggested to include a damping term on the middle pendulum. However, this may not change the modal analysis significantly and can be quite complex to implement.

#### laird HT

HELP!

I am currently working on the derivation of the equations of motion for three coupled pendula, The mass and length of each pendulum is the same, but the central pendulum has some sort of resistive degenerative force due to submersion in a liquid. I have calculated the normal modes without the resistance of the system using the lagrangian approach.

How can i do this, with the resistance integrated into the system?
Any help would be greatly appreciated!

If you have the three equations of motion without the damping force, simply re--write them to include a damping term acting on the middle pendulum.

This is not going to change your modal analysis of the system, however, assuming that you include viscous damping. There is no simple way to include viscous damping in a modal analysis (it can be done using complex modes, but that is a can of worms!), so you are right back where you started as far as the modal analysis is concerned.

Degenerating force in Lagrangian mechanics refers to a resistive force that acts against the motion of a system, causing it to dissipate energy and eventually come to a stop. In your case, it seems that the central pendulum is experiencing this type of force due to its submersion in a liquid.

To incorporate this degenerating force into your system, you will need to modify the Lagrangian equations of motion to include the resistive force term. This can be done by adding a term in the Lagrangian function that represents the energy dissipated by the resistive force. This term will depend on the velocity of the pendulum, as well as the properties of the liquid such as its viscosity.

You can then use the modified Lagrangian equations to derive the equations of motion for your system, taking into account the degenerating force. This will allow you to study the behavior of the system with the resistive force included.

I would recommend consulting with a mentor or colleague who has experience with Lagrangian mechanics to help you with this derivation. Additionally, there are many resources available online that can provide guidance and examples for incorporating degenerating forces into Lagrangian mechanics.

Overall, incorporating the degenerating force into your system will provide a more realistic and accurate representation of the behavior of the pendulum system. I wish you the best of luck in your research.

## 1. What is degenerating force in Lagrangian mechanics?

Degenerating force is a term used in Lagrangian mechanics to describe a force that is dependent on the velocity and position of a system, rather than just the position. This can result in a more complex equation of motion, as the force is not constant and can change as the system moves.

## 2. How does degenerating force affect the equations of motion?

Degenerating force can significantly affect the equations of motion in Lagrangian mechanics, as it adds an extra term to the equation that is dependent on the velocity. This can make the equations more difficult to solve, but also allows for a more accurate representation of the system.

## 3. What are some examples of degenerating force in real-world systems?

Some examples of degenerating forces in real-world systems include air resistance, friction, and resistance from moving through a fluid. These forces are dependent on both the velocity and position of the system, and can greatly impact the motion and behavior of the system.

## 4. How can degenerating force be accounted for in Lagrangian mechanics?

In order to account for degenerating force in Lagrangian mechanics, the equations of motion must be modified to include the extra term that represents the force. This can be done by using the Euler-Lagrange equations, which take into account both the position and velocity dependencies of the force.

## 5. What are the advantages of using Lagrangian mechanics with degenerating force?

One of the main advantages of using Lagrangian mechanics with degenerating force is that it allows for a more accurate and comprehensive description of the system. It also takes into account the effects of external forces, making it a more realistic approach to studying and analyzing physical systems.