Potential Energy

Potential energy is energy which results from position or configuration. The SI unit for energy is the joule = newton x meter in accordance with the basic definition of energy as the capacity for doing work. An object may have the capacity for doing work as a result of its position in a gravitational field (gravitational potential energy), an electric field (electric potential energy), or a magnetic field (magnetic potential energy). It may have elastic potential energy as a result of a stretched spring or other elastic deformation.

Potential Energy Function

If a force acting on an object is a function of position only, it is said to be a conservative force, and it can be represented by a potential energy function which for a one-dimensional case satisfies the derivative condition

The integral form of this relationship is

which can be taken as a definition of potential energy. Note that there is an arbitrary constant of integration in that definition, showing that any constant can be added to the potential energy. Practically, this means that you can set the zero of potential energy at any point which is convenient.

Potential Energy Concept

The potential energy U is equal to the work you must do to move an object from the U=0 reference point to the position r. The reference point at which you assign the value U=0 is arbitrary, so may be chosen for convenience, like choosing the origin of a coordinate system. 势能 U 等于将物体从 U=0 的参考点移动到位置 r 所必须做的功。你指定 U=0 的参考点是任意的，因此可以为了方便而选择，就像选择坐标系的原点一样。 The force on an object is the negative of the derivative of the potential function U. This means it is the negative of the slope of the potential energy curve. Plots of potential functions are valuable aids to visualizing the change of the force in a given region of space. 作用在物体上的力是势函数U的负导数。这意味着它是势能曲线的斜率的负数。势函数的图像对于可视化某一空间区域内的力的变化具有重要作用。

Negative Signs in Potential

F in the definition of potential energy is the force exerted by the force field, e.g., gravity, spring force, etc. The potential energy U is equal to the work you must do against that force to move an object from the U=0 reference point to the position r. The force you must exert to move it must be equal but oppositely directed, and that is the source of the negative sign. The force exerted by the force field always tends toward lower energy and will act to reduce the potential energy. 定义中提到的势能F是力场中作用的力，例如重力、弹簧力等。势能U等于你必须对抗该力将物体从U=0参考点移动到位置r所做的功。你必须施加的力必须大小相等但方向相反，这也是负号的来源。力场中作用的力总是趋向于降低能量，并会作用于减少势能。 The negative sign on the derivative shows that if the potential U increases with increasing r, the force will tend to move it toward smaller r to decrease the potential energy. 导数上的负号表明，如果势能U随r的增加而增加，力将倾向于将其向较小的r方向移动，以减少势能。

Potential Energy Derivative

If the potential energy function U is known, the force at any point can be obtained by taking the derivative of the potential.

Potential Energy Integral

If the force is known, and is a conservative force, then the potential energy can be obtained by integrating the force.

Conservative Force

A conservative force may be defined as one for which the work done in moving between two points A and B is independent of the path taken between the two points. The implication of "conservative" in this context is that you could move it from A to B by one path and return to A by another path with no net loss of energy - any closed return path to A takes net zero work.

A further implication is that the energy of an object which is subject only to that conservative force is dependent upon its position and not upon the path by which it reached that position. This makes it possible to define a potential energy function which depends upon position only.