Dissipative ForceDissipative Force

Non-Potential ForceNon-Potential Force

Generalized force came from Generalized force came from a transformation.a transformation.• Jacobian transformationJacobian transformation

• Not a constraintNot a constraint

Conservative forces were Conservative forces were separated in the Lagrangian.separated in the Lagrangian.

m

i

im q

xFQ

mmmmmQ

q

VQ

q

T

q

T

dt

d

Velocity DependentVelocity Dependent

A function A function MM may exist that may exist that still permits a Lagrangian.still permits a Lagrangian.• Requires force to remain Requires force to remain

after Lagrange equationafter Lagrange equation• MM is a generalized potential is a generalized potential

The Lagrangian must include The Lagrangian must include this to permit solutions by this to permit solutions by the usual equation.the usual equation.

),( jj qqM

jjj q

M

q

M

dt

dQ

MVTL

0

jj q

L

q

L

dt

d

Electromagnetic PotentialElectromagnetic Potential

The electromagnetic force The electromagnetic force depends on velocity.depends on velocity.• Manifold is TManifold is TQQ

Both Both EE and and BB derive from derive from potentials potentials , , AA..• Generalized potential Generalized potential MM

Use this in a Lagrangian, Use this in a Lagrangian, test to see that it returns test to see that it returns usual result.usual result.

)]([ BvEqF

t

AE

vAqqM

AB

vAqqmvL

221

Electromagnetic Electromagnetic LagrangianLagrangian

)()(x

Az

x

Ay

x

Axq

dt

dA

xqxm zyxx

z

A

dt

dz

y

A

dt

dy

x

A

dt

dx

t

A

dt

dA xxxxx

x

L

x

L

dt

d

matching Newtonian equation

xAqqxmL 2

21

)()(z

Az

x

Az

y

Ay

x

Ayq

t

A

xqxm xzxyx

xx BvqqExm )(

Dissipative ForceDissipative Force

Dissipative forces can’t be Dissipative forces can’t be treated with a generalized treated with a generalized potential.potential.• Potential forces in Potential forces in LL

• Non-potential forces to the Non-potential forces to the rightright

Friction is a non-potential Friction is a non-potential force.force.• Linear in velocityLinear in velocity

• Could be derived from a Could be derived from a velocity potentialvelocity potential

jjjQ

q

L

q

L

dt

d

xxfx vbF

221

xxvbF

xx

L

x

L

dt

d

F

Rayleigh FunctionRayleigh Function

imijj qtqbQ ),(

jiij qqb

21F

Dissipative forces can be Dissipative forces can be treated if they are linear in treated if they are linear in velocity.velocity.

This is the Rayleigh This is the Rayleigh dissipation function.dissipation function.

Lagrange’s equations then Lagrange’s equations then include dissipative force.include dissipative force.

0

jjj qq

L

q

L

dt

dF

Energy LostEnergy Lost

The Rayleigh function is related to the 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

dtvFrdFdW fff

dtvbvbvbdW zzyyxxf )( 222

dtdW f F2

Damped OscillatorDamped Oscillator

ExampleExample The 1-D damped harmonic The 1-D damped harmonic

oscillator has linear velocity oscillator has linear velocity dependence.dependence.• Rayleigh function from Rayleigh function from

dampingdamping

• The power lost from The power lost from RayleighRayleigh

Use damped oscillator Use damped oscillator solution to compare with solution to compare with time.time.

next

xbkxF 2

21 xb F

22 xbP F

Em

b

dt

dE

2212

21 kxxmE