How to Solve Cartesian Co-ordinates in Lagrangian Mechanics?

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SUMMARY

This discussion focuses on solving Cartesian coordinates in Lagrangian mechanics, specifically for two scenarios: a plastic knife flying through the air and a piece of chalk rolling inside a stationary cardboard roll. Participants emphasize the need to define Cartesian coordinates that specify the positions of moving masses and to express these coordinates as functions of generalized coordinates. Key insights include the understanding that generalized coordinates encompass both position and momentum, and that the choice of reference frame can simplify the representation of the system.

PREREQUISITES
  • Understanding of Lagrangian mechanics principles
  • Familiarity with Cartesian coordinates and their application in physics
  • Knowledge of generalized coordinates and configuration space
  • Basic grasp of degrees of freedom in mechanical systems
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Students and professionals in physics, particularly those studying mechanics, as well as educators seeking to clarify concepts related to Lagrangian mechanics and coordinate systems.

mooberrymarz
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:confused: :confused: :cry: hey ppl! have just started a course in lagrangian mechanics and it hasnt been the easiest thing. Ahemm, Anywayz we have a assigment to hand in I just don't know how to do these questions. You have to
1. Name cartesian co-ordinates which fully specify the position of all the moving masses in the system.
2. Obtain expressions giving each of the cartesian co-ordinates as functions of the genralised co-ordinates.

]Could any of you show me how to do these 2 and I will try do the rest by myself.]

a. A plastic knife flying through the air
b.A piece of chalk rolling without slipping inside a stationary cardboard roll, but with slippage allowed between the chalk and the board.

Thanx. moo
 
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mooberrymarz said:
:confused: :confused: :cry: hey ppl! have just started a course in lagrangian mechanics and it hasnt been the easiest thing. Ahemm, Anywayz we have a assigment to hand in I just don't know how to do these questions. You have to
1. Name cartesian co-ordinates which fully specify the position of all the moving masses in the system.
2. Obtain expressions giving each of the cartesian co-ordinates as functions of the genralised co-ordinates.

Be more specific, what do you mean by cartesian co-ordinates? Explain the situation clearly, and exactly what your assignment is, its hard to help when only a couple of questions are known from an entire assignment.

a. A plastic knife flying through the air
b.A piece of chalk rolling without slipping inside a stationary cardboard roll, but with slippage allowed between the chalk and the board.

Is this the situation that the two questions apply to?
 
I typed out exactly what my lecturere gave us. You should do 1.) and 2.) for a) and b). In other words for the two cases propose generalised coordinates that directly express the system's position using the minimum amount of parameters. I don't really get what he wants myself.
 
I think mooberrymarz means you got to choose a referenceframe in which the coordinates are as simple as possible.

But what about them generalized coordinates, huh ?


regards
marlon
 
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For a "plastic knife", that is, a rigid body with a length much greater than the other two dimensions, you could do this: choose a point at one end and record its (x, y, z) coordinates. To find the coordinates of any other point, you will need to know its distance from that recorded point as well as the angles the line between them makes with the x, y, z axes (Those are not independent so you really only need two angles). That is, this is a 6 dimensional problem. "Generalized" coordinates means that you must include momenta as independent coordinates.
 
As some have said, "cartesian coordinates" are x,y,z coordinates that locate the object. These are coordinates in a "configuration space".

"Generalized coordinates" are another set of "configuration space" coordinates.
For example, it may be that there is a more convenient set of coordinates, or there may be some constraints on the cartesian coordinates.

A useful question to keep in mind is "How many [configuration] degrees of freedom does the system have?"
A particle confined to a circle has one degree of freedom, although one might use two [obviously not independent] cartesian coordinates to describe it.

http://www.sydgram.nsw.edu.au/CollegeSt/extension/lagrangian/lagrang.pdf
(p 26 may be enlightening)
 
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Thanx you guys! : )
 
Hey moo - good luck on the assignment. It's nice to see you around PF again.
 

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