# Lagrangian mechanics - generalised coordinates question

• I
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.

Homework Helper
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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
curiousPep
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.

Homework Helper
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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!

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