Space-Time symmetries and conservations law

Properties of space 1.Three dimensionality 2.Homogeneity 3.Flatness 4.Isotropy

Properties of Time 1.One-dimensionality 2.Homogeneity 3.Isotropy

Homogeneity of space and Newton third law of motion x’ a y Y’ x z’ z ss’ o o’ x1x1 x2x2 12

Consider two interacting particles 1 and 2 lying along x-axis of frame s Let x 1 and x 2 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(x 1,x 2 ) Let s’ be another frame of reference displaced with respect to s by a distance a along x-axis then oo’= a The principal of homogeneity demands that U(x 1,x 2 ) = U(x’ 1,x’ 2 ) Applying Taylor’s theorem, we get F 12 = -F 21 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. F 12 = -F 21 Newton’s 2 nd law of motion m 1 dv 1 /dt = F (1) m 2 dv 2 /dt = F (2) Adding (1) and (2) and simplifying we get, m 1 v 1 + m 2 v 2 = constant

Isotropy of space and angular momentum conservation x’ x z’ z y’ y dΩdΩ

Let U = U(r 1,r 2 ) be the P.E. of interaction in frame s U = U(r 1 +dr 1,r 2 +dr 2 ) be the potential energy in frame s’. Then using property of isotropy of space U(r 1,r 2 ) = U(r 1 +dr 1,r 2 +dr 2 ) 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.

Homogeneity of time and energy conservation

2 2 2