Exploring Newton's Equations of Motion in Different Coordinate Systems

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Discussion Overview

The discussion centers on Newton's equations of motion for a single particle, specifically how these equations can be expressed in different coordinate systems: Cartesian, cylindrical, and spherical. The scope includes theoretical aspects and practical applications of these equations in various coordinate frameworks.

Discussion Character

  • Technical explanation
  • Exploratory
  • Homework-related

Main Points Raised

  • One participant seeks to understand Newton's equations of motion in Cartesian, cylindrical, and spherical coordinates, starting from the basic form F=ma.
  • Another participant suggests that F=ma can be expressed in vector form across different coordinate systems, emphasizing the need to express the components of force and acceleration in those coordinates.
  • A participant expresses a desire to find the final form of the equations to compare with Hamilton's Equation, indicating a search for a specific result.
  • One participant shares a resource found online that provides the general form for acceleration in spherical coordinates, suggesting that matching force components should be straightforward if the force is known.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the specific forms of the equations in different coordinate systems, and multiple approaches and resources are discussed without resolving the inquiry.

Contextual Notes

Participants reference the need for specific forms of equations and the challenge of expressing acceleration in different coordinate systems, indicating potential gaps in understanding or available resources.

Who May Find This Useful

This discussion may be useful for students or individuals interested in classical mechanics, particularly those exploring the application of Newton's laws in various coordinate systems.

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What are the Newton's equations of motion of a single particle (using cartesian, cylindrical, and spherical coordinates)?

I know the first: F=ma (Along x, y, z) But the in other coordinates?
 
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If you learn about "vectors", you will find that F=ma, written in vector form, is the correct expression for the law in any of those coordinates. You end up just needing to know how to express the components of F and a (the two vectors there) in those different coordinate systems, which is kind of a detail-- it sounds like you really only want the law, and the point of vectors is to able to write the law in just one way. One-stop shopping! The forms for the components in the different coordinates is probably best to just look up, I'll bet a google with the appropriate words would succeed easily. (F is easy, if you understand the coordinates, but a is a bit trickier.)
 
I just need the last form, the final one, so I can see if I reached the same form using Hamilton's Equation. I googled before posting this, I got nothing.
 
Well, all I did was google "acceleration in spherical coordinates", and the first hit was http://www.csupomona.edu/~ajm/materials/delsph.pdf , so if you scroll down to "velocity and acceleration" you find the general form for acceleration in spherical coordinates. Then just match up the force components (radial, latitudinal, longitudinal) which should be easy enough if you know what force you are dealing with, and you have F=ma in component form in spherical coordinates.
 
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Thank you Ken G.
 

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