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Is it possible to derive laws of Newtonian gravity from some symmetries?
Meir Achuz said:It can be derived from Gauss's law and spherical symmetry, if that's what you mean.
Meir Achuz said:You just have to assume spherical symmetry, including that [itex]\vec g[/itex] is in the radial direction.
Then, [itex]\int\int{\vec g}\cdot{\vec{dA}}=-g\int\int dA=-4\pi R^2=-4\pi Gm[/itex].
Meir Achuz said:Why do you say that? Gauss's law is [itex]\int\int{\vec g}\cdot{\vec{dA}}=-4\pi Gm[/itex], with no mention of Newton's law.
Symmetry is the idea that certain physical laws and phenomena remain unchanged or invariant under certain transformations. In the case of Newtonian gravity, this refers to the symmetry of space and time, meaning that the laws of gravity are the same regardless of where or when they are observed.
By assuming the symmetry of space and time, we can use mathematical equations and principles to derive the laws of gravity. This is known as the principle of least action, which states that the path an object takes between two points is the one that minimizes the action (or energy) required. This leads to the equations of motion for objects under the influence of gravity.
The key equations used are Newton's second law of motion (F=ma), the law of universal gravitation (F=Gm1m2/r^2), and the principle of least action (S=∫Ldt). These equations, along with the assumption of symmetry, can be used to derive the equations of motion for objects under the influence of gravity.
Yes, there are some limitations to this approach. For example, it does not take into account the effects of relativity or quantum mechanics, which are necessary for understanding gravity on a larger scale. Additionally, it only applies to objects with mass and does not account for more complex phenomena such as dark matter.
Symmetry is a fundamental concept in many areas of science and physics. It is used to explain and understand various phenomena, from the behavior of subatomic particles to the structure of the universe. It also plays a crucial role in the development of theories and laws, such as the laws of thermodynamics and conservation of energy.