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In the absence of external work, and heat transfer, the total energy of a system (mechanical + thermal) will remain constant. Such a system is commonly referred to as isolated.

Internal conservative forces cannot change the total energy of a system by definition, since the work they do is encapsulated by internal potential energy terms. However, we cannot treat internal non-conservative forces as easily. Though it is clear that for the total energy of the system to be conserved, the total work done by all internal non-conservative forces must be zero.

Is there a way of proving mathematically that this is so, perhaps in the case of a system containing multiple interacting particles? Evidently, internal non-conservative forces can each do non-zero work, and can change the relative amounts of different forms of energy in the system, but they cannot change the total amount.

I wondered whether anyone could help!

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  • $\begingroup$ Look at this example, you have two forces one is depending the first one is spring force the second one is damper force which depended on the velocity. The equation of motion is $\dfrac{d^{2}x}{dt^{2}}=-f\left( x\right) -g\left( \dfrac{dx}{at}\right) $ if multiply this equation with $\dot{x}$ and integrate you get $T+V=-\int \dfrac{dx}{dt}g\left( \dfrac{dx}{dt}\right) dt$ where $\dot{x}$ is the solution of the equation of motion, in the case E=T+V is not constant $\endgroup$
    – Eli
    Apr 3, 2020 at 9:46
  • $\begingroup$ @Eli But in that case, $E$ is the total energy of the spring-body system, and $-g(\frac{dx}{dt})$ is an external force to that system, from perhaps a viscous medium. External forces (conservative or non-conservative) can definitely change the total energy of a system. Crucially, the internal force $-f(x)$ cannot, but it can alter the proportions of $T$ and $V$. $\endgroup$
    – 13509
    Apr 3, 2020 at 9:49
  • $\begingroup$ With I see, but for sure the total energy is not conserved because the damper force, this is what I tried to proof $\endgroup$
    – Eli
    Apr 3, 2020 at 9:57

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The fundamental forces are conservative, so any non-conservative forces are due to neglected degrees of freedom or degrees of freedom we've averaged over. The energy stored in those hidden degrees of freedom is the internal energy, kinda by definition. So internal non-conservative forces must conserve the total energy, internal + mechanical. This is the "statistical mechanics" explanation.

Alternatively, you could take it as an empirical fact. Abstracted from statistical mechanics, the first law is just an empirically derived law - there is in fact a conversion between calories and mechanical joules so that the sum total is conserved.

I'm not sure you could do anything more mathematical. Because indeed, if you allow for non-conservative forces which don't arise from lower-level conservative forces, you're just not going to be able to conserve energy.

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    $\begingroup$ This is a really great explanation, I also forgot to consider than non-conservative forces are actually conservative under the microscope. Thank you! $\endgroup$
    – 13509
    Apr 2, 2020 at 21:53

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