Mechanics
Physics 151
Lecture 3
Lagrange’s Equations
(Goldstein Chapter 1)
Hamilton’s Principle
(Chapter 2)
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Mechanics Langrange's Equation , Lecture Notes - Physics, Study notes of Mechanics
Mechanics, Physics, Lagrange’s Equations, Hamilton’s Principle, D’Alembert’s Principle, Constraint force, Transformation functions, Kinetic energy, Potential energy, harmonic oscillato,r Velocity-Dependent Potential, EM Force on Particle, Monogenic System, Configuration Space, Action Integral , Infinitesimal Path Difference.
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MechanicsPhysics 151
Lecture 3
Lagrange’s Equations(Goldstein Chapter 1)Hamilton’s Principle
(Chapter 2)
What We Did Last Time!^ Discussed multi-particle systems^!^ Internal and external forces
!^ Laws of action and reaction
!^ Introduced constraints^!^ Generalized coordinates!^ Introduced Lagrange’s Equations^!^ ... and didn’t do the derivation "^ Let’s pick it up and start from there
Lagrange’s Equations!^ Express
L = T – V
in terms of generalized coordinates
, their time-derivatives
, and time
t
!^ The potential
V = V
(q,^ t) must exist
!^ i.e. all forces must be conservative
j^
j d^ L
L
dt^ q ^ ∂^ ∂−^ q
^ ^ ∂^
!^
( ,^ , )L q q t^
!T^ V≡^ −
Kinetic energyPotential energy Lagrangian
Recipe
{^ }q^ j
!{ }qj
Virtual Displacement!^ Consider a system with constraints^!^ Ordinary coordinates
r (i^ = 1...i^
N)
!^ Generalized coordinates
q(j^ = 1...j^
n)
!^ Imagine moving all the particlesslightly^!^ Note that
δ r must satisfy the constraintsi^
1 1
1 2 2 2
(^ ,^ ,...,^1 21
(^ ,^ ,...,
(^ ,^ ,...,
n n , ) N^ N^
n q^ q^
q^ t q^ q^
q^ t q^ q^
q^ t = r r = r r ^ " = r r
i^ i^
δi →^ + r r^
r
Virtual displacement i i^
j j^ j
q q
=^ ∑^ ∂
r r 3 N^ coordinatesnot independent
n^ coordinatesindependent j^ j^
j q^ q
q^ δ→ +
D’Alembert’s Principle^!^ Force of constraints dropped out because^!^ Called D’Alembert’s Principle (1743)!^ Now we switch from
r to^ i^
q^ j
!^ Unit of
Q^ not always [force]j^ !^ Q^ qj^
(^ )( is always [work]j
)^
a i i^
i i
δ− = F^ ∑
! p^ r
(^0) δ =i i f^ r
1st term
i i^
j^
j^ j
i^ j^
j q^ j
Q^ q δ q
δ ∂ =^
∑^ ∑
∑ r F^
i j^
i i j Q^
∂ q ≡^
rF ∑ ∂ Generalized force
“constraint” force is out of the game.
You can forget (a)
D’Alembert’s Principle^!^ A bit of work can show!^ D’Alembert’s Principle becomes
,
2nd term
i^
i
i^ i^
i^
j^
i^ i^
j
i^
i^ j^
i j j^
j q^
m^
q
q^
q
δ
δ
δ
∂^
=^
=^
∂^
∑^
∑^ ∑
∑ r^
r
p^ r^
p^
r
!^
!^
!!^2
2 2
i^
i^
i
i j^
j^
j v^
v
d q^ dt
q^
q ^
^
∂^
∂^
→^
^
^
∂^
∂^
^
^
^
r !! (^) r
! j
j^
j^
j d^ T
T
q dt^ q
q
^
^ ∂^
^
=^
^
^ ∂^
^
^
∑^
j^ j
j^
j^
j d^ T
T
Q^
q
dt^ q
q
^
^
^ ∂^
^
−^
−^
^
^
^ ∂^
^
^
^
^
∑^
(^2) mv i (^2) i T^ ≡^ ∑
Lagrange’s Equations!^ Assume that
(^ V^ does not depend on
)^0 j^
T^ V j d^ T dt^ q^
q ^
∂^
∂^ −
^ ^ ∂^
!^ q^ j
V ∂^0 =!q∂ j
0 j^
j d^ L
L
dt^ q ^ ∂^ ∂−^ q
^ ^ ∂^
Finally
(^ ,^
, )^
(^ , )
j^ j^
j
L^ T q
q^ t^
V q^ t =^
Done!
Assumptions We Made!^ Constraints are holonomic^!^ We always assume this!^ Constraint forces do no work^!^ Forget frictions!^ Applied forces are conservative^!^ Lagrange’s Eqn. itself is OK if
V^ depends explicitly on
t
!^ Potential
V^ does not depend on
(^ ,^ ,...,^1
i^ i^
n q^ q^
q^ t
= r r^0 δ^ = f r i i V= −∇ F i i
!^ q^ j
V ∂ 0 =!q∂ j
Will review the last assumption later
Example: Time-Dependent^!^ Transformation functions:^!^ Kinetic energy^!^ Potential energy
(^ ) cos x l^ r^ (^ ) sin
t y^ l^
r^ α tα =^ + =^ + {^
}^ {
}
2 2
2
m^
m T^
x^ y^
r^ l^
r^ α
=^
+^ =
+^ +
!^!^
K^22
V^
r= {^
} 2
2 2
2 (^ ) 2
m^
K
L^
r^ l^
r^
r α =^
+^ +^
2 (^ )
d^ L^
L^ mr
m^
l^ r^
Kr
dt^ r
r
∂^
^ ^ −
=^
−^
+^ +^
^ ∂^
^ ^
Lagrange’s Equation
Example: Time-Dependent^!^ If
K^ >^ m
2 α, a harmonic oscillator with
!^ Center of oscillation is shifted by! If^ K^ <^ m
2 α, moves away exponentially
!^ If^ K
2 = m α
, velocity is constant !^ Centripetal force balances with the spring force
2 (^ )
d^ L^
L^ mr
m^
l^ r^
Kr
dt^ r
r
∂^
^ ^ −
=^
−^
+^ +^
^ ∂^
^ ^
2 2
2
(^
)^
m^ l
mr^ K
m^
r^ K^
α m α^
α ^
+^ −
−^
^
^
2
K^ m^ α m
ω^
Assumptions We Made!^ Constraints are holonomic^!^ We always assume this!^ Constraint forces do no work^!^ Forget frictions!^ Applied forces are conservative^!^ Lagrange’s Eqn. itself is OK if
V^ depends explicitly on
t
!^ Potential
V^ does not depend on
(^ ,^ ,...,^1
i^ i^
n q^ q^
q^ t
= r r^0 δ^ = f r i i V= −∇ F i i
!^ q^ j
V ∂ 0 =!q∂ j
Let’s review the last assumption
Velocity-Dependent Potential!^ We assumed
and^
so that
!^ We could do the same if we had
V ∂^0 =!q∂ j j j^
j d^ T
T
Q
dt^ q ^ ∂^ ∂−^ q
^ ^ ∂^
(^
)^ (
j^
j
d^ T
V^
T^ V
dt^
q^
q ^
∂^ −^
∂^ −−
^
^
∂^
^
This had to be 0
j
j^
j U^ d
U
Q^
q^ dt
^ q ∂^
= −^
+^ ^
∂^
!^ ∂
(^ ,^
, )j j U^ U q
! q^ t=
Generalized,or velocity-dependent“potential”
(^ ,^
, )^
(^ ,^
j^ j^
j^ j
L^ T q
q^ t^
U q^ q
t
=^
!^ −
j
V j Q^
∂= − q∂
Monogenic System!^ If all forces in a system are derived from a generalizedpotential,its called a monogenic system^!^ U
is a function of! Lorentz force is monogenic
!^ A monogenic system is conservative only if^!^ Or!^ Lagrange’s Equation works on a monogenic system
j
j^
j U^ d
U
Q^
q^ dt
^ q ∂^
= −^
+^ ^
∂^
! , ,q q t
U^ U q=
U^ U q^
t ∂^
Hamilton’s Principle!^ We derived Lagrange’s Eqn from Newton’s Eqn usinga “differential principle”^!^ D’Alembert’s principle uses infinitesimal displacements!^ It’s possible to do it with an “integral principle”