Physical interpretation of equations

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SUMMARY

The discussion centers on the physical interpretation of complex mathematical equations in physics, such as v² = u² + 2ax and E=mc². Participants highlight the challenge of deriving intuitive meanings from these equations, especially when squared terms are involved. The conversation references notable figures like Stephen Hawking and Jacob Beckenstein, who had differing views on black hole entropy, illustrating the ongoing debates in the field. Additionally, the discussion touches on the evolution of mathematical concepts, such as the Euler beta function, which later contributed to theories like string theory and the understanding of black holes.

PREREQUISITES
  • Understanding of basic physics concepts, including kinematics and energy-mass equivalence.
  • Familiarity with advanced mathematical functions, such as the Euler beta function.
  • Knowledge of general relativity and black hole physics.
  • Awareness of theoretical physics debates, particularly regarding entropy and quantum mechanics.
NEXT STEPS
  • Explore the physical implications of the Schrödinger Wave equation in quantum mechanics.
  • Study the mathematical foundations of general relativity, focusing on Schwarzschild and Kerr solutions.
  • Investigate the role of the Euler beta function in modern physics, particularly in string theory.
  • Examine the ongoing debates in theoretical physics regarding black hole entropy and the contributions of Hawking and Beckenstein.
USEFUL FOR

This discussion is beneficial for physicists, mathematicians, and students interested in the intersection of mathematics and physical theories, particularly those exploring advanced topics in theoretical physics and cosmology.

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Can all formulas be thought of intuitively/physically?

For example, average speed is change in distance over change in time, that is intuitive and can easily be derived

But when you have something like v^2 = u^2 + 2ax, or e=mc^2, how do you think of it intuitively? Specifically the squared part, what exactly does velocity and the speed of light squared mean physically?
 
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sure,sooner or later. Sometimes via multiple interpretations...like the Schrödinger Wave equation...it's meaning is STILL debated...

but it may take quite a while to get to that stage (of physical interpretation) since as you suggest some math is so complicated it's physical meaning may take years to determine. And different people may have opposing/contradictory views...like Hawking and Jacob Beckenstein regarding the entropy of black holes...Hawking (as I recall) thought Beckenstein was nutty...until he arrived at the same conclusion via a different mathematical approach.

And typically mathematicians are not all that interested in the physical meaning when they may develop some new math...often they develop math and physicists come along and use it in a physical situation. When some math appears we may not have any theory to go with it.

One example is the Euler beta function which was later seen to be an aspect of the strong nuclear force...and even caused people to stumble intio string theory! String theory started out as one subject and someone later realized the math included gravitons...and so it became what it is today. Another example is some of the mathematics black holes: frames of reference...and the solution(s) to Einstein's equation of General Relativty...Einstein had the equation, Schwarszchild, Kerr, to name two, figured out solutions (for non rotating and rotating black holes) ...and what they meant...what the equation said physically about black holes.

Similar situation regarding
 
I do not have a good working knowledge of physics yet. I tried to piece this together but after researching this, I couldn’t figure out the correct laws of physics to combine to develop a formula to answer this question. Ex. 1 - A moving object impacts a static object at a constant velocity. Ex. 2 - A moving object impacts a static object at the same velocity but is accelerating at the moment of impact. Assuming the mass of the objects is the same and the velocity at the moment of impact...

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