1. Dissipative Force

2. Non-Potential Force  Generalized force came from a transformation. Jacobian transformationJacobian transformation Not a constraintNot a constraint  Conservative forces were separated in the Lagrangian.

3. Velocity Dependent  A function M may exist that still permits a Lagrangian. Requires force to remain after Lagrange equationRequires force to remain after Lagrange equation M is a generalized potentialM is a generalized potential  The Lagrangian must include this to permit solutions by the usual equation.

4. Electromagnetic Potential  The electromagnetic force depends on velocity. Manifold is T Q  Both E and B derive from potentials , A. Generalized potential M  Use this in a Lagrangian, test to see that it returns usual result.

5. Electromagnetic Lagrangian matching Newtonian equation

6. Dissipative Force  Dissipative forces can’t be treated with a generalized potential. Potential forces in LPotential forces in L Non-potential forces to the rightNon-potential forces to the right  Friction is a non-potential force. Linear in velocityLinear in velocity Could be derived from a velocity potentialCould be derived from a velocity potential

7. Rayleigh Function  Dissipative forces can be treated if they are linear in velocity.  This is the Rayleigh dissipation function.  Lagrange’s equations then include dissipative force.

8. Energy Lost  The Rayleigh function is related to the energy lost. Work done is related to powerWork done is related to power Power is twice the Rayleigh functionPower is twice the Rayleigh function

9. Damped Oscillator Example  The 1-D damped harmonic oscillator has linear velocity dependence. Rayleigh function from dampingRayleigh function from damping The power lost from RayleighThe power lost from Rayleigh  Use damped oscillator solution to compare with time. next