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Homogeneity of Time

Time is homogeneous means one instant of time (or duration) is identical to any other instant (or duration) of time. An experiment, whether performed today or tomorrow or a year later will yield same result under otherwise same conditions.

Let us discuss the effect of concept of homogeneity of time on Newton’s laws of motion. Homogeneity of time implies that the forces in nature should not depend explicitly on time. [For example, the force F of gravitation between the two point masses m₁ and m₂ separated by distance r remains the same at time t (today) and t’ (tomorrow).]

That is, for all kinds of forces, we have

Now, if a force is conservative, we can define a potential energy of interaction U such that the force is negative gradient of U, i.e. we have

This means that if F does not depend explicitly on t, then equivalently, potential energy U also is not an explicit function of t, or,

U = U (x, y, z)

where, x, y, z refer to co-ordinates of the particle under consideration in a given inertial frame.

The explicit independence of U on t leads to conservation of mechanical energy of the particle. To show this, let us write the total energy of the particles as,

E = 1/2 mv² + U

Hence, we get

= v . F + (–F_x v_x – F_y v_y – F_z v_z)

= 0

or, E remains constant in time. Thus the law of conservation of energy; in Newtonian mechanics, follows directly as a consequence of homogeneity of time.

Homogeneity of time implies that we can choose the zero of time at any instant for the observation of a physical process.