Kinetic and Potential Energy Definitions in Physics
Kinetic Energy1
When the force depends only on the position, i.e., it is independent of the velocity or time, the equation of motion (differential equation) for the straight-line motion of a particle is as follows.
$$ \begin{equation} F(x)=m\ddot{x} \label{force1} \end{equation} $$
In this case, acceleration $\ddot{x}$ can be expressed in terms of velocity as follows.
$$ \begin{align*} \ddot{x} &= \dfrac{d \dot{x}}{dt} \\ &=\dfrac{dv}{dt} \\ &=\dfrac{dv}{dx} \dfrac{dx}{dt} \\ &=v\dfrac{dv}{dx} \\ &= \frac{1}{2}\frac{ d (v^{2})}{ dx } \end{align*} $$
Substituting this into $(1)$,
$$ F(x)=m\ddot{x}= m\frac{1}{2}\frac{d(v^{2})}{dx}=\frac{ d }{ dx }\left( \frac{1}{2}mv^{2} \right) $$
The physical quantity inside the parentheses in the above equation is defined as the particle’s kinetic energy and is denoted by $T$.
$$ T=\dfrac{1}{2}mv^2 $$
Then, the equation of motion $(1)$ can be expressed as follows.
$$ F(x)=\dfrac{dT}{dx} $$
The symbol for kinetic energy is often written as $K$ or $E_{K}$, following the first letter of kinetic.
Potential Energy
Now, let’s define a function $V(x)$ as follows.
$$ -\dfrac{dV(x)}{dx}=F(x) $$
The function $V(x)$ defined as above is called potential energy2. Then, it can be written as follows, like the kinetic energy.
$$ -\frac{ d V(x)}{ d x}=F(x)=\frac{ d T}{ d x} $$
Integrating the above equation from the initial position $x_{0}$ to the later position $x_{1}$ results in the following.
$$ -V(x_{1}) +V(x_{0}) =T_{1}-T_{0} $$
What this equation means is that during an object’s motion, the change in potential energy is the same in magnitude but opposite in sign to the change in kinetic energy. That is, if one increases, the other decreases by an equal amount. This implies that their sum is always constant. Therefore, let’s denote the sum of the two as the particle’s total energy or mechanical energy and mark it as $E$.
$$ E=T_{0}+V(x_{0})=T_{1}+V_{x_{1}} $$
This equation is called the energy equation. As seen above, if the force can be obtained from a position-dependent function, the potential energy $V(x)$, the mechanical energy of the particle is conserved; therefore, that force is called a conservative force. In cases where the force is not a conservative force, i.e., when there is no position-dependent potential energy, it is called a nonconservative force. When a nonconservative force acts on an object, the mechanical energy of the object is not conserved.