Time flows uniformly i.e., it is homogeneous in nature. If we perform an experiment and change the time of the experiment then the result of that experiment remains unchanged. This indicates that the result of an experiment is independent of the change in the origin of time. This phenomenon of time is known as homogeneity.

Laws of conservation of energy from the homogeneity of time:

Let, \( K \) be the kinetic energy and \( U \) be the potential energy of a system. So the total energy of this system is given by,

\( E=U+K \).

Let us consider two masses \( m_1 \) and \( m_2 \) are kept at a distance \( \vec{r} \) apart. Then the gravitational force between them is given by,

\( \vec{F}_G=G\frac{m_1\ m_2}{r^2}\hat{r}\tag{1} \)

where, \( G \) is the gravitational constant.

Again, let us consider two charged particles of charges \( q_1 \) and \( q_2 \) are kept at a distance \( \vec{r} \) apart, then the coulomb force between them is given by,

\( \vec{F}_C=\displaystyle{\frac{1}{4\pi{\epsilon}_0}\frac{q_1\ q_2}{r^2}}\hat{r}\tag{2} \)

where, \( \displaystyle{\frac{1}{4\pi{\epsilon}_0}} \) is a constant and \( \epsilon_0 \) is the permittivity of free space.

From equations (1) and (2) we can say that time does not appear explicitly in the expressions of force.

We know that for a conservative system,

\( \vec{F}=-\displaystyle{\frac{\partial{U}}{\partial{r}}}\hat{r} \)

So \( U \) is function of \( \vec{F} \) and \( \vec{r} \). Since the time flows uniformly, so the force acting on particle does not depend on time, this means that the force \( \vec{F} \) is a function of \( \vec{r} \) only. Again \( U \) is a function of \( \vec{F} \) and \( \vec{r} \), so it is the only function of \( \vec{r} \). Thus according to the principle of homogeneity of time, we can write,

\( \displaystyle{\frac{\partial{U}}{\partial{t}}}=0\tag{3} \)

Again, if a particle of mass \( m \) moves with velocity \( v \), then the kinetic energy of the particle is \( K=\displaystyle{\frac{1}{2}}mv^2 \). So we clearly say that the kinetic energy also does not depend on time.

Therefore, \( \displaystyle{\frac{\partial{K}}{\partial{t}}}=0\tag{4} \)

If the total energy \( E \) is a function of \( \vec{r} \) and \( t \), then we can write,

\( E=E(\vec{r},t) \)

\( or,\ dE=\displaystyle{\frac{\partial{E}}{\partial{r}}dr+\frac{\partial{E}}{\partial{t}}dt} \)

\( or,\ dE=\displaystyle{\frac{\partial{(U+K)}}{\partial{r}}dr+\frac{\partial{(U+K)}}{\partial{t}}dt} \)

\( or,\ dE=\displaystyle{\left(\frac{\partial{U}}{\partial{r}}+\frac{\partial{K}}{\partial{r}}\right)dr+\left(\frac{\partial{U}}{\partial{t}}+\frac{\partial{K}}{\partial{t}}\right)dt} \)

Now using equations (3) and (4), we get,

\( dE=\displaystyle{\left[\frac{\partial{U}}{\partial{r}}+\frac{\partial{K}}{\partial{r}}\right]dr} \)

\( or,\ \displaystyle{\frac{dE}{dt}=\left[\frac{\partial{U}}{\partial{r}}+\frac{\partial{K}}{\partial{r}}\right]\frac{dr}{dt}}\tag{5} \)

Now,

\( \displaystyle{\frac{\partial{K}}{\partial{r}}=\frac{\partial}{\partial{r}}\left(\frac{1}{2}mv^2\right)=mv\frac{\partial{v}}{\partial{r}}=mv\frac{dv}{dr}} \)

[since, v does not depend on time explicitly]

\( or,\ \displaystyle{\frac{\partial{K}}{\partial{r}}=m\frac{dr}{dt}\frac{dv}{dr}} \)

\( or,\ \displaystyle{\frac{\partial{K}}{\partial{r}}=m\frac{dv}{dt}} \)

\( or,\ \displaystyle{\frac{\partial{K}}{\partial{r}}=ma} \)

[where, \( \displaystyle{a=\frac{dv}{dt}} \) is the acceleration]

And we know, \( \displaystyle{\frac{\partial{U}}{\partial{r}}=-F} \)

So putting these values in equation (5), we get

\( \displaystyle{\frac{dE}{dt}=(-F+ma)\frac{dr}{dt}}\tag{6} \)

According to Newton’s second law of motion, \( F=ma \), so from equation (6), we get

\( \displaystyle{\frac{dE}{dt}=0} \)

\( or,\ \displaystyle{E=constant}. \)

Now it is proved that homogeneity of time and Newton’s Second law of motion lead to the principle of conservation of energy.