Recently in my university lectures I happened to understand the reasons for existence of conservative laws such as law of conservation of momentum, angular momentum, energy and others.
Later, on further discussion and Research, I understood the following point:
Law of conservation of Momentum arises due to homogeneity of space and translation of symmetry of universe.
Can someone please brief out what exactly is "Homogeneity of Space"?
The way I understand it, homogeneity of Space means that the Lagrangian possess translational Invariance i.e. doesn't change when you do a transformation of the type: \begin{equation} \tilde{x}^{i}=x^{i}+\epsilon^{i} \end{equation}
Then from Noether's Theorem, you can derive the following:
\begin{equation} \begin{aligned} 0 &=\delta S \\ &=\sum_{i} \int_{t_{1}}^{t_{2}}\left(\frac{\partial L}{\partial x^{i}} \delta x^{i}+\frac{\partial L}{\partial \dot{x}^{i}} \delta \dot{x}^{i}\right) d t \\ &=\sum_{i} \int_{t_{1}}^{t_{2}}\left(\frac{\partial L}{\partial x^{i}} \delta x^{i}+\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{x}^{i}} \delta x^{i}\right)-\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{x}^{i}}\right) \delta x^{i}\right) d t \\ &=\left.\sum_{i} \frac{\partial L}{\partial \dot{x}^{i}} \varepsilon^{i}\right|_{t_{1}} ^{t_{2}}+\sum_{i} \int_{t_{1}}^{t_{2}}\left(\frac{\partial L}{\partial x^{i}}-\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{x}^{i}}\right)\right) \varepsilon^{i} d t \end{aligned} \end{equation}
Taken that your particle obeys the Euler-Lagrange equation the integral vanishes. Since $t_1,t_2$ and $\epsilon^i$ are all arbitrary we can conclude that:
\begin{equation} \frac{\partial L}{\partial \dot{x}^i}=0 \end{equation}
but that's just the momentum of the particle, therefore momentum is conserved.
I'm gonna try to also give an intuitive explanation. Let's say you throw a stone off a cliff. Gravity makes each height unique and the motion of the stone will depend on the height of the cliff you chose. If you try to do the same experiment on a nearby cliff with the same height you will get the exact same result as the previous one. Now think of the stone, just like you wouldn't be able to distinguish one cliff from another, the stone can't distinguish nearby horizontal points but it can distinguish nearby vertical points. We know from Newton's equation that if you do the experiment, momentum changes in the vertical direction, the case where nearby points are distinguishable for the stone. Therefore we expect that in the horizontal direction, where nearby points are indistinguishable, momentum is conserved.
According to this website
homogeneous space has the same properties at every point, it is uniform without irregularities
In simpler terms: In a carefully defined system, mechanical energy is conserved if there are no forces which originate from outside of the system and no dissi0pative forces (like friction or collisions) within the system. (The work done by an external force may be included in an energy calculation.) Momentum is conserved in the absence of external forces, and angular momentum is conserved in the absence of external torques.
The easiest way I know to describe this is as follows:
Imagine that you have a cool laboratory in which you can explore the properties of the universe, that is a trailer that you can tow to any location, park, and start doing physics.
What you notice is that the fundamental laws of physics you derive do not depend at all on where you might happen to park the trailer. This means (via Noether's Theorem) that linear momentum is conserved.
Next, you rotate that trailer so it points in various directions in 3-D space and repeat your experiments. You find that the results are the same no matter which way the trailer lab is oriented in 3-D space and again via Noether's Theorem, you discover that angular momentum is conserved.
Finally, you repeat the whole series of experiments in one place and in one direction, and you discover that the results of your experiments do not depend on whether you perform those experiments on a Tuesday or a Thursday. Noether's theorem then tells you that kinetic energy is a conserved quantity.
