Galilean Invariance and constraints on Forces.

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Discussion Overview

The discussion centers on the constraints imposed on forces between two interacting point particles under the assumptions of isotropy and homogeneity of space and time. Participants explore whether Newton's third law can be derived in this context and examine the implications for conservation laws, particularly momentum.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants propose that under isotropy and homogeneity, constraints on forces can be derived, potentially leading to a derivation of Newton's third law.
  • Others argue that Newton's third law does not hold in all cases, such as electromagnetic interactions, where the momentum of the field must be considered for conservation of momentum.
  • A later reply emphasizes that in the general case of position and velocity dependent forces, conservation of linear and angular momentum can be maintained, referencing Noether's theorem.
  • One participant reiterates the initial question about the constraints on forces, suggesting a mathematical representation of the forces as a function of the relative positions of the particles.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of Newton's third law in various contexts, particularly in relation to electromagnetic interactions. The discussion remains unresolved regarding the specific constraints on forces under the stated assumptions.

Contextual Notes

The discussion includes assumptions about isotropy and homogeneity that may not be universally applicable. There is also a lack of consensus on the derivation of Newton's third law and the role of field momentum in conservation laws.

andresB
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Let's say we have a system of two point particles that can interact with each other by forces that are position and velocity dependent. The forces might or might not be derivable from a generalized potential.
Assuming Isotropy of space and homogeneity of space and time, what are the constraints imposed on the possible forces between the particles? In particular, can Newton's third law be "derived" under such conditions?
 
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andresB said:
Assuming Isotropy of space and homogeneity of space and time, what are the constraints imposed on the possible forces between the particles? In particular, can Newton's third law be "derived" under such conditions?
Newton’s third law generalizes to the conservation of momentum and per Noether’s theorem the conservation of momentum is a consequence of the homogeneity of space.
 
Indeed.

But, for example, in the case of electromagnetic interaction there is no Newton third law and you need to consider the momentum of the field to have conservation of momentum.

In the most general case of position and velocity dependent forces, what can be said about the forces if we assume homogenity and isotropy of space?
 
andresB said:
But, for example, in the case of electromagnetic interaction there is no Newton third law and you need to consider the momentum of the field to have conservation of momentum.
Hence the word “generalizes” in my response above.

andresB said:
In the most general case of position and velocity dependent forces, what can be said about the forces if we assume homogenity and isotropy of space?
They will conserve linear momentum and angular momentum. Per Noether’s theorem.
 
andresB said:
Let's say we have a system of two point particles that can interact with each other by forces that are position and velocity dependent. The forces might or might not be derivable from a generalized potential.
Assuming Isotropy of space and homogeneity of space and time, what are the constraints imposed on the possible forces between the particles?
##\boldsymbol F=\boldsymbol F(\boldsymbol r_1-\boldsymbol r_2)##
 
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