Physics 319 Classical Mechanics G. A. Krafft Old Dominion University Jefferson Lab Lecture 7 G. A. Krafft Jefferson Lab
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
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.
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
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
Demonstration Total Energy Conserved If several conservative forces are acting on a particle, the same argument applies to show is conserved.
Relation Between Force and Potential In general Because true in any direction
Gradient Operation Gradient of a function defined as Gradient “operator” Useful general relation Function changes most quickly in the direction of gradient
Path Independence Curl Operation Quantifies how much the field rotates (curls) locally Path independence follows from Stokes Theorem Vanishing of curl guarantees path independence
Path Independence of Coulomb Force Coulomb force is curl-free (conservative)