BEZOUT IDENTITIES WITH INEQUALITY CONSTRAINTS

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STATIC EQUILIBRIUM
4 Calkin, M. G. Lagrangian and Hamiltonian
Mechanics, World Scientific, Singapore,
1996, ISBN 981-02-2672-1
Consider an object having mass M held in the air
at height y gt 0 above the ground and released.
Energy conservation implies that its height y(t)
as a function of time t gt 0 since release must
satisfy the equation
Since this equation holds if and only if either
energy conservation doesnt imply Newtons 2nd law
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STATIC EQUILIBRIUM
We observe that
and, for an unconstrained object having mass M,
position r (a vector), and a position dependent
potential energy U(r), we observe that energy
conservation implies that
An object is said to be in static equilibrium if
its position r is constant. Therefore, if a
system is not in static equilibrium then energy
conservation implies Newtons 2nd law.
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ENERGY NOT ENOUGH
We made the painful observation, that
conservation of energy by itself does not suffice
to explain the real world in which things happen
rain falls from the sky and objects move when
they are pushed
Definition A virtual displacement of an
object is any change of its position that can be
imagined without violating any specific
constraint. The virtual work
is the scalar product of the force
and the virtual displacement
Examples Virtual displacements for a falling
object are arbitrary vectors, but those for an
object that is constrained to lie on a table
must be horizontal
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STATIC PRINCIPLE OF VIRTUAL WORK
Fundamental Principle An object is in static
equilibrium only if the virtual work associated
with every virtual displacement is zero
Example 1 Consider an unconstrained object that
is subjected to a force F. Then, if the object is
in static equilibrium, we obtain from the
principle above that
for every virtual displacement
Therefore, since
the object is unconstrained we may choose
to obtain that
hence
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STATIC PRINCIPLE OF VIRTUAL WORK
Example 2 Consider an object that is constrained
to lie on a table surface that is flat. Assume
that the only force on the object is the
gravitational force F. Then if the object is in
static equilibrium
for every virtual displacement
Since the
object is constrained to lie on the table
surface, the virtual displacements consist of all
vectors parallel to the table. Therefore, if the
object is in static equilibrium F must be
orthogonal to the table surface. (Recall that two
vectors are orthogonal if and only if their
scalar product equals zero)
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STATIC PRINCIPLE OF VIRTUAL WORK
Example 3 This example shows that the static
principle of virtual work applies to systems
consisting of one or more objects that may be
mutually as well as individually constrained.
Consider two weights attached by a flexible rope
as illustrated below.
gravitational force
Let the configuration of the system be
represented by a vector
whose coordinates are the heights of each object.
If the system is in static equilibrium what is
the ratio of the weights ?
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STATIC PRINCIPLE OF VIRTUAL WORK
Example 4 Consider two weights on a pulley as
illustrated in the figure below
gravitational force
If the system is in static equilibrium what is
the ratio of the weights ?
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DALEMBERTS PRINCIPLE
Definition For a moving object the force of
constraint is
DAlemberts Principle The virtual work done by
force of constraint equals zero, this means that
for every virtual displacement
This is a dynamic principle of virtual work. It
can be extended to describe systems with many
objects.
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DALEMBERTS PRINCIPLE
Example 1 An object slides on an inclined plane
(x, y -horizontal coordinates) having angle
and DAlemberts principle implies that
therefore the equations of motion are
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GENERALIZED COORDINATES
The configuration of a system with N objects
having masses
whose positions
are constrained by 3N-f independent functions
can be parameterized by f independent variables
Virtual displacements, expressed in GC, are
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LAGRANGES EQUATIONS
DAlemberts principle expressed in GC has the
form
Therefore, the chain rule for derivatives implies
that
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LAGRANGES EQUATIONS
DAlemberts Principle implies, but can NOT be
derived from Newtons 2nd Law. However, it can be
derived from conservation of energy together with
the static principle of virtual work by defining
constraints as the limit of forces orthogonal to
the constraint set.
If each applied force is conservative then there
exists a potential energy function
such that
This is the reason
are called generalized forces.
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LAGRANGES EQUATIONS
Example 1 Consider a single unconstrained object
with mass M and position r. We can choose q r
to obtain
Therefore, Lagranges Equations reduce to
Newtons Second Law
This is the reason
are called
generalized momenta.
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LAGRANGES EQUATIONS
Example 2 Compute the trajectory of an object
with mass M sliding on a surface whose height
h(x,y) (x and y are horizontal coordinates)
Choose generalized coordinates
Then the kinetic energy
And the generalized momenta
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LAGRANGES EQUATIONS
Example 2 For i1,2 the generalized force
and Lagranges Equations
are a pair of second order differential equations.
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LAGRANGES EQUATIONS
Example 2 A tedious, but direct computation yields
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LAGRANGES EQUATIONS
Example 2 where A is the 2 x 3 matrix with entries
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LAGRANGES EQUATIONS
Example 2
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GEODESICS AND GENERAL RELATIVITY
If we set g 0 in Example 2 we obtain the
equations for a geodesic trajectory on the
surface, that is the uniform speed trajectory
whose distance between any two points is minimal.
Albert Einstein showed that an object in any
inertial frame, such as in space moving with
constant speed or falling freely under the Earths
gravity, follows a geodesic trajectory. This is
how he explained the equivalence between
gravitational and inertial mass that was measured
by experimentalists with such amazing precision.
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HAMILTONS EQUATIONS AND ENERGY
If, as in Example 2, the kinetic energy is a
quadratic function of the generalized velocities
and the forces are conservative (arising from a
potential energy U) then the total energy is
given by the Hamiltonian
The system dynamics satisfies Hamiltons equation
the foundation of advanced and quantum mechanics.