I am self-studying the classical mechanics using the book by Taylor, and I have a question about the potential energy.

The book (pg 111) says:

If all forces on an object are conservative, we can define a quantity called the potential energy PE, denoted by $U(r)$, a function only of position, with the property that the total mechanical energy $$E = KE + PE = T + U(r) $$ is constant.

Then, the book goes on to say that we can define the potential energy $U(r)$ corresponding to a given conservative force.

Am I correct to say that for each conservative force $F$, there is a potential energy $U_{F}(r)$ and if all forces are conservative, then the sum of the corresponding potential energies and the kinetic energy is constant?


Yes you are.

If a force is conservative, its work does not depend of any path between any points $A$ and $B$. Since the work integral can depend only on the initial and final points themselves, we define $$W_{A\rightarrow B}=\int_A^B\vec F\cdot d\vec r\equiv U(A)-U(B).$$ Now define the mechanical energy as $E=K+U$ so that $$dE=dK+dU.$$ Suppose there are two forces on the particle, a conservative $\vec F$ and a non conservative $\vec F_{nc}$. The energy-work theorem asserts that $dK=dW=(\vec F+\vec F_{nc})\cdot d\vec r$ while the potential energy variation is $dU=-\vec F\cdot d\vec r$. Therefore $$dE=\vec F_{nc}\cdot d\vec r,$$ and the mechanical energy is conserved as long as there is no non conservatives forces in the system.


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