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I know that energy conservation is related to time-translation invariance, but I don't understand how to check this invariance and what it means.

I have an example case, that confuses me:

Assume you are holding a bowling ball and you are standing still on ice. (no friction as usual)
This means both you and the ball have no kinetic energy.
If you now throw the bowling ball both you and the ball will have kinetic energy, that means energy is not conserved, but only momentum is conserved.

Can you add a potential energy term to the total energy, so that energy will be conserved in such a case?
This would have to be a potential energy that is a step function regarding the time.
Would such a potential energy be possible / would it still conserve energy?

Or is this kind of potential (discontinuous in time) the definition of non-time-translation-invariance?

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2 Answers 2

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Not sure this is answering your question, but here's my thoughts laid out in this expression:

Let $V= K_1 + K_2$ where $K_1$ and $K_2$ are the kinetic energies of you and the ball respectively. $V_1$ is the energy stored up in your muscles to just execute that function of throwing the ball. $V$ is the potential energy of the ball.

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In order to throw the ball you need to extend your muscles. Extending your muscles uses chemical energy. Once you add up all the sources of energy - kinetic energy of yourself and the ball, chemical energy stored in your muscles, energy that is "lost" because it goes into heat - then you find the energy is conserved.

If you want to throw the ball without using the chemical energy in your muscles, then you need to find some other source of energy. This could be the potential energy in a compressed spring, or in the stretched elastic that powers a catapult, or the potential energy of a raised weight that powers a trebuchet, or the chemical energy released in an explosion that fires the ball from a cannon, or the electrical energy that powers a railgun. In all cases, energy is conserved once you account for all the sources of energy.

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