Constraints and Generalized Coordinates Video Lecture - Civil Engineering (CE)

in this video I want to talk about
constraints and generalized coordinates
and we'll use this example of a pendulum
that we've already solved to sort of see
the ideas and and I'll link them to the
more formal language you might see in a
textbook so what did we start with we
started out with this pendulum that's in
two dimensions right it has it can move
in the x direction and it can move in
the y direction right it swings
side-to-side and in the process it goes
up and down so one way we could have
solved this is we could have found
equations for x and y that had
derivatives of x and y and then we could
solve those differential equations to
find the equations of motion and x and y
and then put them together to get the
total motion of this pendulum right and
that would have been kind of hard right
this isn't this isn't a very easy
problem to do an X and y coordinates and
rectangular or Cartesian coordinates but
we said hey we know that this pendulum
is going to move along a circular path
right it's going to stay a fixed
distance from this point here where they
the string is tied and a convenient
thing to do would be to use polar
coordinates right will have a fixed
distance L and this angle theta will
change so it'll just move in this circle
and now we only have one variable to use
and that actually makes things a lot
simpler so what did we do here we
started out with two degrees of freedom
right two degrees of freedom
I'll just write it out two degrees of
freedom and then we used some of our
knowledge right we know that this thing
is going to move along a circular path
so we use our knowledge and that
knowledge is called the constraint this
this lives in a two-dimensional space
but it's constrained by the fact that
it's moving in a circular path because
the string isn't stretching or bouncing
or anything and smoothly sewing it back
and forth so we had two degrees of
freedom and we had one constraint one
constraint right and we have an equation
for this constraint right it's the
equation of a circle so we have the
at l-squared actually use green since
it's in green green here l squared
equals x squared plus y squared right
that's our equation of constraint so
I'll write it here equation EQ and
equation of constraint actually I should
probably use a different word here I
shouldn't say degrees of freedom here I
should say not degrees of freedom two
dimensions two dimensions right the
vertical direction in the horizontal
direction and then we have two
dimensions so without any constraints
that would be two degrees of freedom but
with the constraint there's only one
degree of freedom so one degree of
freedom
one degree of freedom so all of this is
just a long way of saying that we can
just use theta to specify the motion of
this pendulum just one degree of freedom
we only need one variable one
differential equation for that variable
and that's all we need so to be more
general actually let's cross this out to
so so this whole thing I wrote initially
is crossed out and say it's actually
three dimensions three dimensions
because in general things can move in
three dimensions so we can write down a
sort of general equation that says s and
s is the degrees of freedom degrees of
freedom degrees of freedom DOF degrees
of freedom equals three three four three
dimensions so number of dimensions
number of dimensions
and then I'll write n 3n n is the number
of particles we're looking at so here we
only looked at this one mass on the end
of this string but maybe there are two
pendulums running around or maybe it's a
gas with many many many atoms in it in
that case n would be very large but but
each object can move in three dimensions
so we'll need three times the number of
particles variables to specify the
motions of the entire system
so that's what this 3n is three
dimensions for each particle but from
that we can subtract our number of
constraint equations and it will use the
letter M here M is the number of
constraint equations so constraints and
here our constraint equation was this
the equation that says we know this
thing will be on a circle write the
equation of the circle so for this
example we have one pendulum and and we
could say actually we could say we have
since we said this is in three
dimensions right we can make another
constraint equation and say that
a Z equals zero right and this this
equation of constraint is just saying
that we know this thing is just going to
stay in a plane it's just a
two-dimensional motion so we have three
dimensions generally in space but we're
saying that one of them doesn't change
so we can set it to zero or you can set
it to any constant but now we have two
equations of constraints so two
equations of constraint hopefully all
this crossing out isn't too confusing so
we had three dimensions one object two
equations of constraint and so we end up
with one degree of freedom so we only
needed one coordinate theta to specify
the motion of this pendulum so in this
case theta takes the role of our
generalized coordinate since s is one
here there's one degree of freedom we
only need one generalized coordinate so
theta is our generalized coordinate and
we're not always going to be so lucky
that there's only one generalized
coordinate is our generalized coordinate
and I guess I should add some arrows to
this just to try to make it more clear
number of particles number of particles
right so we'll do some examples where
where things are a little more
complicated than this and then we'll use
this to sort of figure out how many
generalized coordinates we need and then
we'll use our equations of constraint to
find those general s coordinates and
then once we do that we will well be in
good shape for the rest of the problem
and we can just apply our lagrangian and
put it in be in the Euler Lagrange
formula and get our differential
equations of motion
you