# Galilean Invariance and constraints on Forces.

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## Main Question or Discussion Point

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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Dale
Mentor
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?

Dale
Mentor
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.

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.

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)##

• Ibix and Dale