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11. Hamilton's principle
LAST
TIMES
- We isolated a generic class of variational problems.
- The goal was to find a function z(x) for which the
integral
over some property
L(z,dz/dx,x)
had an extremum.
- We derived the Euler-Lagrange equation
for the solution to the problem.
- We analyzed, as an example, the brachistochrone problem and Fermat's
principle.
PRINCIPLE OF LEAST ACTION
(or Hamilton's
principle)
The Euler-Lagrange equation for the variational principle
and the Lagrangian equation of motion derived in lecture 8 for a conservative system are of the same form. To
see this let the independent variable x represent time and the dependent
variable z be a generalized coordinate. This suggests that Newtonian
mechanics can be derived from a variational principle. We now reformulate the
mechanics of conservative systems by requiring that they are characterized by a
Lagrangian which is a function of the generalized coordinates and
velocities of the particles constituting the system. The actual trajectory is
then the one for which the action
is extremal subject to boundary conditions at the endpoints. The
integration is with respect to time. Usually the action will be a minimum hence
the name principle of least action.
Logically we consider this to be a
postulate taking the place of Newton's laws as the foundation of mechanics. It
then remains to construct the Lagrangian -something which, of course, depends on
the problem at hand. Also, we are not saying that Newton was wrong- we want the
Lagrangian formulation to reproduce Newton's law when applicable.
Caveat: I mention in passing that the actual path is not always a
minimum for the entire path, but only for sufficiently short segments. This is
of no problem in practice. In deriving the equation of motion we only use the
extremum condition.
The Euler-Lagrange equation is linear in the
Lagrangian. We can multiply
by a constant
without changing the equation of motion. We choose the Lagrangian to have
dimension of energy.
Example:
Particle in one
dimension subject to velocity independent force:
We put
v=dx/dt. If the Lagrangian is
the equation of motion is
We write
for the kinetic energy
where a is the acceleration. Similarly if V(x) is
the potential energy
If we put
we see that equation of motion becomes the familiar
f=ma
GENERALIZED COORDINATES
In our previous example x was
the Cartesian coordinate of the particle. It need not be, as discussed lecture 7 we may use any set of generalized coordinates which
involves imposing holonomic constraints on Cartesian coordinates.
MANY
DEGREES OF FREEDOM
Most often we are dealing with systems requiring a number
of generalized coordinates to describe the motion.
Suppose N
coordinates are required to specify the motion (after we have substituted for
the holonomic constraints). We say that the system has N degrees of
freedom.
The variational principle is now
We can carry out the variation independently for each of the coordinates
and obtain a set of N Euler-Lagrange equations
i.e. one equation for each coordinate.
Example
PENDULUM WITH MOVABLE SUPPORT
A pendulum of length r,
mass m. Its support has mass M and it can slide without friction
horizontally (coordinate X). The horizontal and vertical components of
the pendulum mass are
The velocity components are
Hence the Lagrangian is
![\begin{displaymath}{\cal L=T-V}\end{displaymath}\begin{displaymath}=\frac{M}{2}\...
...\dot{\theta}^2
+2\dot{X}\dot{\theta}r\cos\theta]+mgr\cos\theta\end{displaymath}](hamilton's_files/img17.gif) |
(1) |
We will come back to the equations of motion for this system later.
GENERALIZED FORCES AND MOMENTA.
When the kinetic energy is on the
form
and
is velocity
independent, the equation of motion can be written
where p is the momentum and f the force
In the general case:
generalized
momentum
In lecture 7 we defined the generalized conservative force as
the partial derivative of the potential energy with respect to the generalized
coordinate. We now modify the definition so that
generalized
force
The Lagrangian equations of motion can thus be written:
If the Lagrangian does not depend explicitly on one of the
coordinates the corresponding generalized force is zero and the corresponding
generalized momentum is conserved!
Example THE PENDULUM
The generalized force is
Physically the generalized force associated with the angle
is the torque!
The generalized momentum is
which we recognize as the angular momentum. The Lagrangian equation of
motion is thus just
Rate of change of angular momentum=torque
Example
PENDULUM WITH MOVABLE SUPPORT
The Lagrangian (1)
doesn't depend explicitly on X hence
is conserved. A bit of reflection will convince you that this is just the
equation for the conservation of linear momentum in the x-direction.
So there is nothing new!
We could have obtained the above results
without resorting to Lagrangians.
However, if the system is complicated the
Lagrangian approach offers the possibility of proceeding in a systematic
fashion, without having to worry about free body diagrams, normal forces, or
pseudo forces due to acceleration of coordinate system.
The systematic,
algebraic, approach makes it much easier to avoid errors!
SUMMARY
We
have
- developed Lagrangian dynamics from the principle of least action
- shown that for a conservative system with velocity-independent forces we
could reproduce Newtonian dynamics if we put for the Lagrangian
- introduced generalized momenta
- modified the definition of generalized force
- shown that if Lagrangian did not depend on a generalized coordinate the
corresponding momentum was conserved.
Example problems:
Problems 1 and 3 of 2000
midterm , with solution.
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Birger Bergersen
2001-01-31