Is it possible to apply the law of conservation of mechanical energy in all reference frames, including non-inertial frames, when there is no frictional or drag force? My personal belief is that it is only applicable in inertial frames, but I am open to being proven wrong. If there are counterarguments or evidence to support its applicability in non-inertial frames, I kindly request an explanation and demonstration.
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$\begingroup$ Where is the figure mentioned in your question? You can post your diagram by editing the post and selecting the square potraight icon. $\endgroup$– Duke WilliamCommented Jun 28, 2023 at 16:20
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$\begingroup$ Include your conceptual concern. Perhaps the difference in kinetic energy is not the same as the difference in gravitational potential energy? $\endgroup$– JEBCommented Jun 28, 2023 at 16:32
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$\begingroup$ I got two equations from using the conservation of energy and the conservation of momentum by taking the final velocities of the wedge and block v1, v2 respectively. But I can't solve for the difference in kinetic energy? $\endgroup$– Thasarath KandathasanCommented Jun 29, 2023 at 3:23
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$\begingroup$ The first sentence is incorrect. Energy is conserved as long as there aren't any non-conservative force that work, it doesn't matter that the frame is inertial or not. There are even cases where some inertial forces (the centrifugal force) becomes conservative. $\endgroup$– MiyaseCommented Jun 29, 2023 at 7:57
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$\begingroup$ Good edits. I have voted to reopen. It is now a conceptual question instead of a homework question. Hopefully one more person will agree. $\endgroup$– DaleCommented Jun 29, 2023 at 17:32
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1 Answer
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One of the most influential theorems in all of physics is Noether's theorem. She proved that conserved quantities come from some sort of differential symmetry of the corresponding action (assuming of course that the laws of physics can be formulated as an action principle).
In the case of conservation of energy, the corresponding symmetry is time symmetry. In other words, there is a conserved energy if the Lagrangian is symmetric over time, meaning if the laws of physics are the same yesterday, today, and tomorrow, then energy is conserved.
Inertial frames have this property, so energy is conserved in inertial frames. However, non-inertial frames are more varied. Some non-inertial frames will have time symmetry and others will not.
Examples with time symmetry (conserved energy): uniformly accelerating, rotating with constant angular velocity
Examples without time symmetry (no conserved energy): oscillating, accelerating suddenly, rotating with non-constant angular velocity
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$\begingroup$ Can you (briefly) explain why uniformly accelerating reference frames would have energy conservation? I thought in such a frame everything would be accelerating so energy ($\sum\frac{1}{2}mv_{i}^{2}+U$) is not conserved. $\endgroup$ Commented Aug 7, 2023 at 6:34
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$\begingroup$ @MaximalIdeal yes. Noether’s theorem shows that energy is related to the symmetry under time translation. Uniform acceleration is time translation symmetric, so by Noether’s theorem there is a conserved energy. $\endgroup$– DaleCommented Aug 7, 2023 at 11:03