Lagrangian mechanics - generalised coordinates question

• I
• curiousPep
In summary, you need to find the Lagrangian for the (symmetric) spinning top in terms of Euler angles.
curiousPep
I think I undeerstand Lagrangian mechanics but I have a question that will help to clarify some concepts.
Imagine I throw a pencil. For that I have 5 generalised coordinates (x,y,z and 2 rotational).
When I express Kinetic Energy (T) as:
$$T = 1/2m\dot{x^{2}}+1/2m\dot{y^{2}}+1/2m\dot{z^{2}} + I\dot{\theta^{2}} + I\dot{\phi^{2}}$$
and potential energy (V)
$$V=mgz$$
Then I use Lagrangian to find the EOM.
For x,y,z is fine but for $$\theta$$ and $$\phi$$ I have a question. I see how x,y,z can be a expressed as functions of $$\theta\;and\;\phi$$, but why should I do this. I mean in cases that something is less obvious, then I will get the wrong EOM.
Thank you, and I hope the latex code works.

You need double dollars or double hashes to delimit Latex here.

Im not sure I understand your question. A point particle has linear KE. A extended rigid body has linear KE of the CoM plus rotational KE of the body about the CoM. You have to know to consider both.

vanhees71
PeroK said:
You need double dollars or double hashes to delimit Latex here.

Im not sure I understand your question. A point particle has linear KE. A extended rigid body has linear KE of the CoM plus rotational KE of the body about the CoM. You have to know to consider both.
I will try to explain it better, cause I see it's a bit confusing.
When I have a rigid body like a pencil of length 2L, the generalized coordinates defined are x,y,z (COM relative to (0,0)) and
$$\theta, \phi$$ (Euler's angles).
However, x,y,z can be expressed as functions of $$\theta,\phi$$.
For example: $$x = X + Lsin\theta cos\phi$$. My question is that, why do I need to do this in oder to find the right EOM for $$\theta\;and\;\phi$$?
Or is this not needed?I mean in a more complex case the relationship mu not be that obvious, so I won't know if my EOM are right or not.

If ##(X, Y, Z)## are the coordinates of the CoM, then that's what you need for ##T##. The rotational motion involves the moment of inertia ##I##, which encapsulates the position of every point mass in the body in terms of calculating rotational KE.

Your ##x## above seems to be just the position of one end of the pencil!

The right degrees of freedom are the center-of-mass position components ##\vec{X}## and the three Euler angles for the rotation of the rigid body around this center of mass. So what you look for is the Lagrangian for the (symmetric) spinning top in terms of Euler angles. See, e.g.,

https://hepweb.ucsd.edu/ph110b/110b_notes/node36.html

To simplify the equations somewhat, you may neglect the rotation around the axis of the pencil, because the corresponding moment of inertia is small compared to that around any axis perpendicular to it, i.e., you may set ##I^{(3)}=0##.

wrobel

1. What is Lagrangian mechanics?

Lagrangian mechanics is a mathematical framework used to describe the motion of a system of particles or objects. It is based on the principle of least action, which states that the motion of a system is determined by minimizing the difference between its kinetic and potential energy.

2. What are generalised coordinates in Lagrangian mechanics?

Generalised coordinates are a set of independent variables that are used to describe the configuration of a system. They are chosen in such a way that they can uniquely determine the position of each particle in the system. These coordinates can be any variables that are convenient for describing the system, such as position, angle, or velocity.

3. How are generalised coordinates related to traditional Cartesian coordinates?

Generalised coordinates are related to traditional Cartesian coordinates through a mathematical transformation. This transformation allows us to express the position, velocity, and acceleration of a particle in terms of the generalised coordinates. By using generalised coordinates, we can simplify the equations of motion and make them easier to solve.

4. What is the significance of using generalised coordinates in Lagrangian mechanics?

Using generalised coordinates in Lagrangian mechanics allows us to describe the motion of a system in a more natural and convenient way. It also simplifies the equations of motion, making them easier to solve. In addition, generalised coordinates can be chosen based on the symmetries and constraints of the system, which can provide insight into the underlying physics.

5. Can Lagrangian mechanics be applied to any system?

Yes, Lagrangian mechanics can be applied to any system, as long as it satisfies the principle of least action. This includes systems with constraints, such as a pendulum or a rolling ball, as well as more complex systems with multiple particles and forces. Lagrangian mechanics is a powerful tool that has been applied to a wide range of physical systems, from simple mechanical systems to quantum field theory.

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