Extending the Lagrangian of a double pendulum to more complex systems

In summary, the paper discusses the extension of the Lagrangian formulation used for a simple double pendulum to analyze more complex dynamical systems. It emphasizes the importance of incorporating additional degrees of freedom, constraints, and interactions that arise in these systems. The authors provide mathematical techniques for deriving the Lagrangian and highlight potential applications in robotics and engineering, showcasing how this approach can enhance the understanding of complex motion and stability in various mechanical systems.
  • #1
KataruZ98
27
3
The total kinetic energy of a double pendulum can be calculated through the formule reported in the following article: https://dassencio.org/33

This works if the double pendulum in question is formed by two masses connected to each other and — one of them — to the point of origin by a "massless" rod. However, I'm interested in expanding this formula to cover systems where two bodies of more complex shapes are the swinging parts of the "double pendulum", if it possible that is.

Say I have a Cartesian plane, and at the origin point there's a cylinder of height H and radius R with the y-axis passing through the center of mass. On top of it is a cone placed upside down, with the center of mass also passing through the y-axis. Said cone has radius r and height h.

The cylinder is then tipped to the right, forming now an angle θ1 with the x-axis. As a consequence, the cone swings as well, producing an angle θ2 respect to the horizontal. It all happens within time t. How can I work out the Lagrangian from the data given?
 
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  • #2
Gravity would topple the upside down cone and make it fall down from cylinder top.
 
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  • #3
anuttarasammyak said:
Gravity would topple the upside down cone and make it fall down from cylinder top.
Suppose for the sake of the argument that the two are connected, not simply placed one below the other.
 
  • #4
Then we may regard it as double pendulum around its upright position of
[tex]\theta_1,\theta_2=\pi[/tex]forgetting surface contact of cylinder and cone. Cylinder mass center position should be considered for in detail.
 
Last edited:
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FAQ: Extending the Lagrangian of a double pendulum to more complex systems

What is the Lagrangian formulation of mechanics?

The Lagrangian formulation is a reformulation of classical mechanics that uses the principle of least action. It is defined as the difference between the kinetic energy (T) and potential energy (V) of a system, expressed as L = T - V. This approach allows for the analysis of complex systems and can simplify the equations of motion by using generalized coordinates.

How do you extend the Lagrangian of a double pendulum to more complex systems?

To extend the Lagrangian of a double pendulum to more complex systems, you identify the additional degrees of freedom and the corresponding kinetic and potential energies of each component. You then express the total kinetic and potential energies in terms of generalized coordinates and velocities. The new Lagrangian is constructed by subtracting the total potential energy from the total kinetic energy of the system, allowing you to derive the equations of motion for the more complex configuration.

What are generalized coordinates, and why are they important?

Generalized coordinates are variables that describe the configuration of a system in a way that simplifies the analysis of its dynamics. They can be angles, distances, or any other parameters that define the position of the system's components. Generalized coordinates are important because they allow for a more flexible and efficient formulation of the Lagrangian, especially in systems with constraints or non-Cartesian geometries.

What challenges arise when extending the Lagrangian to systems with more degrees of freedom?

When extending the Lagrangian to systems with more degrees of freedom, challenges include increased complexity in the expressions for kinetic and potential energy, as well as the potential for more complicated constraints. Additionally, the resulting equations of motion may become more difficult to solve analytically, necessitating numerical methods or approximations to obtain solutions. Careful consideration of the coupling between different components is also crucial.

How can numerical methods be used to solve the equations of motion for complex systems?

Numerical methods can be used to solve the equations of motion for complex systems by discretizing time and employing algorithms such as the Euler method, Runge-Kutta methods, or symplectic integrators. These methods allow for the simulation of the system's dynamics over time, providing approximate solutions to the equations derived from the Lagrangian. Numerical simulations are particularly useful when analytical solutions are intractable due to the complexity of the system.

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