1. Physics 319 Classical Mechanics
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 7
G. A. Krafft
Jefferson Lab

2. Energy Kinetic energy (energy inherent in movement) Time derivative
For a small displacement in time
where dr is the displacement after dt
Integrating along the orbit gives the Work-Energy theorem
The rhs of this equation: the work going from 1 to 2

3. Line Integrals Integrals of this form, called line integrals,
are most easily handled by converting the integrand to a one-dimensional integral over some parameter.
For a central force
By the change of variables formula from calculus it does not matter how the integral is parameterized, but can depend on the path chosen.

4. Conservative Forces
A force (field) is called “conservative” if and only if
The force depends only on position. It should not depend on velocity, or depend explicitly on time.
The work integral between two points is independent of the path chosen to perform the line integral. (We’ll have a “test” procedure in a few slides!)
For conservative forces
One can define a single function, the potential energy function, by
During a motion governed by this force, the total energy is conserved

5. Examples For a uniform electric field in the x-direction
Near the surface, the gravitational field of the earth is nearly uniform. Defining z = 0 as at the surface
3-D Linear Restoring Force
Spherically symmetric case: all the ks equal

6. Demonstration Total Energy Conserved
If several conservative forces are acting on a particle, the same argument applies to show
is conserved.

7. Relation Between Force and Potential
In general
Because true in any direction

8. Gradient Operation Gradient of a function defined as
Gradient “operator”
Useful general relation
Function changes most quickly in the direction of gradient

9. Path Independence Curl Operation
Quantifies how much the field rotates (curls) locally
Path independence follows from Stokes Theorem
Vanishing of curl guarantees path independence

10. Path Independence of Coulomb Force
Coulomb force is curl-free (conservative)