Space-Time symmetries and conservations law

Properties of space

1. Three dimensionality2. Homogeneity3. Flatness4. Isotropy

Properties of Time

1. One-dimensionality2. Homogeneity3. Isotropy

Homogeneity of space and Newton third law of motion

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o o’x1 x2

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Consider two interacting particles 1 and 2 lying along x-axis of frame sLet x1 and x2 are the distance of the particles from o. the potential energy of interaction U between the particles in frame s is given by U=U(x1,x2)

Let s’ be another frame of reference displaced with respect to s by a distance a along x-axis then oo’= aThe principal of homogeneity demands that U(x1,x2) = U(x’1,x’2)Applying Taylor’s theorem, we get F12 = -F21

This is nothing but Newton’s third law of motion.

Homogeneity of space and law of conservation of linear momentum

• Consider tow interacting particles 1 and 2 of masses m1 and m2 then forces between the particles must satisfy Newton’s third law as required by homogeneity of space .

• F12 = -F21

• Newton’s 2nd law of motion• m1dv1/dt = F12 --------- (1)• m2dv2/dt = F21 ---------(2)• Adding (1) and (2) and simplifying we get ,• m1v1 + m2v2 = constant

Isotropy of space and angular momentum conservation

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dΩ

Let U = U(r1,r2) be the P.E. of interaction in frame sU = U(r1+dr1,r2+dr2) be the potential energy in frame s’. Then using property of isotropy of space U(r1,r2) = U(r1+dr1,r2+dr2)Applying Taylor’s theorem, we get dL/dt = 0L = constant This is just the law of conservation momentum and is a consequence of space.

Homogeneity of time and energy conservation

• Consider tow interacting particles 1 and 2 lying along x-axis of frame s. The P.E. between the two particles is given by

• U = U(x1,x2)• If x1 and x2 change w.r.t. time then U will also change

with time but U is an indirect function of time. The homogeneity of time demands that result of an experiment should not change with time

• =0

Let us assume U = U(x1,x2,t)dU = dtUsing newton’s 2nd law , we get d/dt(1/2m1v21 + 1/2m2v2 + U)=0Or 1/2m1v1 + 1/2m2v2 + U = constantWhich is nothing but law of conservation of totalEnergy.

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