Transcript Document 7750252
Configuration Space
• The state of a system of
n
particles & subject to
m
constraints connecting some of the
3n
rectangular coordinates is completely specified by
s = 3n – m generalized coordinates .
Sometimes its convenient to represent the state of such a system by a point in an abstract
s
-dimensional space called
CONFIGURATION SPACE
. Each dimension in this space corresponds to one of the coordinates
q j
. This point specifies the
CONFIGURATION
of the system at a particular time. As the
q j
change in time (governed by the eqtns of motion) this point traces out a curve in configuration space. The exact curve depends on the initial conditions. Often we speak of “the path” of the system as it “moves” in configuration space. • Obviously, this is
NOT
the same as the particle path as it moves in ordinary 3d space!
Lagrange’s Equations in Generalized Coordinates
Section 7.4
• In light of the preceding discussion we can now restate
Hamilton’s Principle
in slightly different language:
Of all of the possible paths along which a dynamical system may move from one point to another in configuration space within a specified time interval, the ACTUAL PATH followed is that which minimizes the time integral of the Lagrangian for the system.
• Repeat
Hamilton’s Principle & Lagrange Equations
derivation in terms of
generalized coordinates
: • Use the fact that the Lagrangian
L L
is a
SCALAR
.
is invariant under coordinate transformation (Ch. 1). –
Also allowed:
Transformations which change
L
but leave the equations of motion unchanged. For example it can easily be shown that if
L
L + (d[f(q j ,t)]/dt),
this obviously changes
L
but leaves Lagrange Equations of motion unchanged. (Valid for
ANY f(q j ,t)!
).
• Always use the definition
L = T - U
. This is valid in a proper set of generalized coordinates.
L
can also contain an
arbitrary constant
. This comes from the arbitrary constant in PE
U
.
It doesn’t matter if we express L in terms of rectangular coordinates & velocities x α,i generalized coordinates & velocities q j & x & q j .
α,i or L = T(x α,i ) - U(x α,i ) (1)
• The transformation between rectangular & generalized coordinates & velocities is of the form:
x α,i = x α,i (q j ,t); x α,i = x α,i (q j ,q j ,t)
So, we also have
Note!
L = T(q j ,q j ,t) - U(q j ,t) (2)
• The Lagrangians in
(1)
&
(2)
are the same! From
(2),
in generalized coordinates we have
L
L(q j ,q j ,t)
•
Hamilton’s Principle in generalized coordinates:
δ∫L(q
j
,q
j
,t) dt = 0
• This is again
identical
(limits
t 1 < t < t 2
) to the abstract calculus of variations problem of Ch. 6
with the replacements:
δJ
δ∫Ldt, x
t, y i (x)
q j (t) y i
(x)
(dq j (t)/dt) = q j (t) f[y i (x),y i
(x);x]
L(q j ,q j ,t)
The Lagrangian L satisfies Euler’s equations with these replacements!
• These become
Lagrange’s Equations in generalized coordinates:
(
L/
q
j
) - (d/dt)[(
L/
q
j
)] = 0 (A)
• For a system with
s
degrees of freedom and
m
constraints, there are
j = 1,2, ..,s
coupled
equations like
(A)
+
m
constraint equations. Together, these
s +
m equations
give a complete description of the system motion
!
• Note that the
validity of Lagrange’s Equations requires
that: –
All forces
(other than constraint forces)
must be conservative
(that is a PE,
U
must exist).
• However, in graduate mechanics, it is shown how to generalize the Lagrangian formalism to dissipative (non-conservative) forces!
– The equations of constraint must be relations that connect the coordinates of particles & they may be functions of time:
f k (x α,i ,t) = 0, k = 1,2, .. m (B)
• Constraints which may be expressed in the form
(B)
Holonomic Constraints
.
• If the constraint equations,
(B)
, do not explicitly contain the time
Fixed or Scleronomic Constraints
.
• Time dependent constraints
Rhenomic Constraints
.
• Here, we only consider conservative forces & holonomic constraints.
Simple Example 7.3
• Projectile motion (with no air resistance). Find the equations of motion in both Cartesian & polar coordinates.
•
Cartesian: x
= horizontal,
y
= vertical
T = (½)mx 2 + (½)my 2 , U = mgy
L = T - U = (½)m(x 2
Lagrange’s Equations:
+ y 2 ) - mgy (
L/
x) - (d/dt)[(
L/
x)] = 0, (1) (
L/
x) = 0, (
L/
x) = mx, (d/dt)[(
L/
x)] = mx, (1)
x = 0 (
L/
y) - (d/dt)[(
L/
y)] = 0, (2). (
L/
y) = -mg, (
L/
y) = my (d/dt)[(
L/
y)] = my. (2)
mg + my = 0; y = -g
•
Polar: x = r cosθ , y = r sinθ r 2
=
x 2
+
y 2 ; tanθ = (y/x) T = (½)mr 2 + (½)mr 2 θ 2 U = mgr sinθ
L = T- U = (½)m(r 2 + r 2 θ 2 ) - mgr sinθ
Lagrange’s Eqtns:
(
L/
r) - (d/dt)[(
L/
r)] = 0, (3) (
L/
r) = mrθ 2 – mg sinθ, (
L/
r) = mr, (d/dt)[(
L/
r)] = mr (3)
rθ 2 – g sinθ - r = 0 (
L/
θ) - (d/dt)[(
L/
θ)] = 0, (4) (
L/
θ) = - mgr cosθ, (
L/
θ) = mr 2 θ (d/dt)[(
L/
θ)] = m(2rr θ + r 2 θ) (4)
gr cosθ + 2rr θ + r 2 θ = 0
Clearly, its simpler using Cartesian coordinates as the generalized coordinates!
Example 7.4 - Non-trivial!
• A particle, mass
m
, is constrained to move on the inside surface of a smooth cone of half angle
α
(figure). It is subject to gravity. Determine a set of generalized coordinates & determine the constraints. Find Lagrange’s Equations of motion.
Worked on the board!!
m Constraint: z = r cotα
Example 7.5
• See figure. The point of support of a simple pendulum of length
b
moves on a massless rim of radius
a
, rotating with a constant angular velocity
ω
. Obtain the expressions for the Cartesian components of the velocity and the acceleration of the mass
m
.
m (x,y)
Obtain also the angular acceleration for the angle
θ
shown in the figure. Worked on the board!
Example 7.6
• See figure. Find the frequency of
small oscillations
of a simple pendulum placed in a railroad car that has a constant acceleration
a
in the
x
-direction. Worked on the board!!
Note: the railroad car is not an inertial frame!
g
Example 7.7
• A bead slides along a smooth wire bent in the shape of a parabola,
z = cr 2
as in the figure. The bead rotates in a circle or radius
R
when the wire is rotating about its vertical symmetry axis with angular velocity
ω
. Find the value of
c
. Worked on the board!
Example 7.8
• Consider the double pulley system shown in the figure. Use the coordinates indicated and determine the equations of motion. Worked on the board!