Energy Conservation

Consider a particle moving in a conservative force field. Suppose that the particle moves from point $A$ to point $B$ along some particular path. According to Equation (1.35),

$$\int_{{\bf r}_A}^{{\bf r}_B}{\bf f}\cdot d{\bf r} = K_B-K_A.$$ (1.48)

However, Equation (1.41) implies that

$$\int_{{\bf r}_A}^{{\bf r}_B}{\bf f}\cdot d{\bf r} = -\int_{\bf0}^... ...{\bf f}\cdot d{\bf r} + \int_{\bf0}^{{\bf r}_B}{\bf f}\cdot d {\bf r}=U_A -U_B,$$ (1.49)

where $U_A$ is the potential energy at point $A$, et cetera. The previous two equations yield

$$K_A + U_A = K_B+U_B.$$ (1.50)

Thus, if we define the total energy, $E$, of the particle as the sum of its kinetic and potential energies,

$$E = K+ U,$$ (1.51)

then we deduce that $E$ is a constant of the motion. In other words, the total energy of a particle moving in a conservative force field is a conserved quantity.