Mechanics
Physics 151
Lecture 4
Hamilton’s Principle
(Chapter 2)
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Mechanics Hamilton Principle, Lecture Notes - Physics, Study notes of Mechanics
Mechanics, Physics, Hamilton’s Principle, Configuration Space, Action Integral, Infinitesimal Path Difference, Calculus of Variations, Lagrange’s Equation, Notation of Variation, Momentum Conservation, Generalized Momentum, Angular Momentum, Conservation Laws.
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MechanicsPhysics 151
Lecture 4
Hamilton’s Principle(Chapter 2)
Administravia!^ Problem Set #1 due^!
Solutions will be posted on the web after this lecture
!^ Problem Set #2 is here^!
Due next Thursday
!^ Next lecture (Tuesday) will be given by Srinivas andAbdol-Reza^!
I will be attending a workshop at Stanford
Today’s Goals!^ Discuss Hamilton’s Principle^!
Derive Lagrange’s Eqn from Hamilton’s Principle! Calculus of variation! Looks unfamiliar, but not so difficult
!^ Discuss conservation laws again^!
Using Lagrangian formalism! Linear, angular momenta! Connection between symmetry, invariance of theLagrangian, and conservation of generalized momentum
Configuration Space!^ Generalized coordinates
q ,...,^1
q fully describe then^
system’s configuration at any moment! Imagine an
n -dimensional space
!^ Each point in this space (
q ,...,^1
q )n^
corresponds to one configuration of the system! Time evolution of the system
"^ A curve in the
configuration space
configurationspace
real space
configuration space
Hamilton’s Principle!^ This is equivalent to Lagrange’s Equations^!
We will prove this
!^ Three equivalent formulations^!
Newton’s Eqn depends explicitly on
x-y-z
coordinates
!^ Lagrange’s Eqn is same for any generalized coordinates!^ Hamilton’s Principle refers to no coordinates^!
Everything is in the action integral
The action integral of a physical system is
stationary
for the actual path
We will also define “stationary”
Hamilton’s Principle is more fundamental
probably...
Stationary!^ Consider two paths that are close to each other^!
Difference is infinitesimal
!^ Stationary means that thedifference of the action integrals iszero to the 1st order of
δ q ( t )
!^ Similar to “first derivative = 0”! Almost same as saying “minimum”!^ It could as well be maximum
configuration space t^1
t^2 ( ) q t ( )^
q t^
q t δ+
2
2
1
1
(^
,^
, )^
( ,^ , )
t^
t
t^
t
I^
L q^
q q^
q t dt
L q q t dt
δ
δ^
δ
=^
+^
+^
−^
∫^
∫ !^
!^
1
2 ( )^
(^ )^
q t^
q t
δ^
Hamilton
"
Lagrange
!^ Consider 1 generalized coordinate
q
!^ Add
δ q ( t ) to
q ( t ), then make
δ q ( t )
"^0
!^ Do this by^!
α^ is a parameter
"^0
!^ η(
t ) is an arbitrary well-behavingfunction
!^ Let’s define
( )^
q t^
t
δ^
αη=^ Continuous, non-singular,continuous
η' and
η''
(^21) (^ )^
( ( ,^
),^ ( ,
t t I^
L q t
q t
t dt
α^
≡^ ∫^
( ,^ )
q t^
q t^
t
=^
1
2 ( )^
(^ )^
t^
t
η^
configuration space t^1
t^2 ( ) q t ( )^
q t^
q t δ+
NB: this alsodepends on
η( t )
Calculus of Variations!^ Let’s define^!
If the action is stationary
(^21) (^ )^
( ( ,^
),^ ( ,
t t I^
L q t
q t
t dt
α^
=^ ∫^
0 (^ )^
dI d
α α^ α=
^
^ =
^
^
(^21) (^ )^
t t dI^
L dq
L dq
dt
d^
q d^
q d
α^
^
∂^
=^
^
∂^
^
∫^
Some work! t 2 L^ t 1
d^ L
dq
dt
q^ dt
q^
d^ α
^
∂^
=^
∂^
^
∫^
( , ( ) t η=
)^
( )^
q t^
q t^
t
=^
Arbitrary function
for any
η(t)
NB: this alsodepends on
η( t )
Notation of Variation!^ For shorthand, we use
δ^ for infinitesimal variation
!^ I.e.
α-derivative at
α^ = 0
!^ Hamilton’s Principle can be written as
(^21)
tL t
d^
L
I^
qdt q^ dt
q
^
∂^
=^
−^
^
∂^
^
∫^
0
dq q^
d^
t d
d^ α
α^ η
^ ≡ α^ =
^
^
(^
)
(^21) 0
( ( ,^
),^ ( ,
t t
dI^
d
I^
d^
L q t
q t
t dt d
d^
d α
α^
= ^
≡^
^
^
^
∫^
Going Multi-Coordinates!^ Trivial to expand
q^ "
( q ,^1
q , …,^2
q ) n
!^ See Goldstein Section 2.3!^ Assumption:
δ q ,^ δ^1
q , … are arbitrary and independent^2
!^ Not true for
x-y-z
coordinates if there are constraints
!^ True for generalized coordinates if the system isholonomic
(^21)
t
i
t^ i^
i^
i L^ d
L
I^
q dt q^ dt
q
δ
δ
^
∂^
=^
−^
^
∂^
^
∑∫
!^ = 0 for each
i
Calculus of Variation!^ Technique has wider applications^!
In general for! Examples in Goldstein Section 2.2! Most famous: the brachistochrone problem
0 J
δ^ =
dy ′ ≡ y dx
2 (^1
( ),^
( ),^
x x J^
f^ y x
y^ ′ x^ x dx
=^ ∫
f^ d
f y^ dx
^ y
∂^
−^
^
∂^
Fastest path via gravity
Conservation Laws!^ We’ve seen (in Lectures 1&2) conservation of linear,angular momenta and energy in Newtonian mechanics^!
How do they work with Lagrange’s equations?! Should better be the same…
!^ We’ll find a few differences and assumptions^!
They are, in fact, limitations we ignored so far
Generalized Momentum!^ Let’s call
the generalized momentum
!^ Also known as canonical or conjugate momentum!^ Equals to usual momentum for simple
x-y-z
coordinates
!^ Lagrange’s equation becomes^!
p is conserved if j^
L^ does not depend explicitly on
qj
!^ Such
q is called cyclic (or ignorable) j^
j
L j p^
∂≡! q ∂^
j
j dp^
L
dt^
∂− = q ∂
Generalized momentum associatedwith a cyclic coordinate is conserved
Linear momentumconservation is aspecial case
Generalized Momentum!^ Generalized momentum may not look like linearmomentum^!
Dimension may vary, if
q is not a space coordinate j^
!^ pqj
always has the dimension of action (= work j^
×^ time)
!^ Form may vary if
V^ depends on velocity
!^ Example: a particle in EM field
j
L j p^
∂≡! q ∂^
2 (^1) L mv^ 2
q^
q φ
=^
−^
+^
⋅ A v^
x^
x p^
mx^
qA =^
Extra term due to velocity-dependent potential