Conservative force
A force is conservative if it can be written as for some potential ; equivalently, the work done along a trajectory depends only on its endpoints, not on the path itself.
Definition 2.2 (Conservative Force)
Conservativeforces form an important class of forces that can be written as
for some potential (also called potential energy) .
Path independence of work
Proposition 2.6
Conservative forces have the property that the work done by the force as a particle moves along a trajectory , where the work done is defined as
only depends on the endpoints of the trajectory, not on the path itself.
Proof
Let the trajectory go from at to at . Then
using Newton’s second law and conservation of energy. Hence depends only on the endpoints of the trajectory.
Proof (Direct)
Using results from IA Vector Calculus,
Related
Stated in
- Definition 2.2 (Conservative Force)§2.2 Conservative Forces and Gravity
- Proposition 2.6§2.3 Conservation of Energy