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Space-Time symmetries and conservations law

Space-Time symmetries and conservations law. Properties of space . Three dimensionality Homogeneity Flatness Isotropy. Properties of Time. One-dimensionality Homogeneity Isotropy. Homogeneity of space and Newton third law of motion. y. Y’. s. s ’. 1. 2. a. o. o ’. x’. x. x 1.

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Space-Time symmetries and conservations law

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  1. Space-Time symmetries and conservations law

  2. Properties of space • Three dimensionality • Homogeneity • Flatness • Isotropy

  3. Properties of Time • One-dimensionality • Homogeneity • Isotropy

  4. Homogeneity of space and Newton third law of motion y Y’ s s’ 1 2 a o o’ x’ x x1 x2 z z’

  5. Consider two interacting particles 1 and 2 lying along x-axis of frame s Let 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 thenoo’= a The 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.

  6. 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

  7. Isotropy of space and angular momentum conservation y y’ x’ x dΩ z z’

  8. Let U = U(r1,r2) be the P.E. of interaction in frame s U = 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 = 0 L = constant This is just the law of conservation momentum and is a consequence of space.

  9. 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

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

  11. THANK YOU

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