Forest UCM LEq
Lagrange's Foramlism for Classical Mechanics
Hamilton's principle falls out of the calculus of variations in that seeking the shortest time interval is the focus of the variations.
- Of all possible paths along which a dynamical system may move from on point to another within a specified time interval, the actual path followed is that which minimizes the time integral of the difference between the kinetic and potential energies.
Casting this in the language of the calculus of variations
if you want the above "action" integral to be stationary then according to the calculus of variations you want the Euler-Lagrange equation to be satisfied where
- the generalized force if I have conservative forces
- Newton's second law in an Inertial reference frame = time derivative of the generalized momentum
Lagrange's Equations in generalized coordinates
Generalized coordinatesare a set of parameters that uniquely specify the instantaneous state of a dynamical system.
The number of independent generalized coordinatesis given by subtracting the number of constraints from the number of degrees of freedom .
Consider the 2-D pendulum where an object of massis constrained by a rod of length . The object is at one end of the rod and the rod is fixed to rotate about the other end.
- There are 32 degrees of freedom for the 2-D problem
- The particle is constrained to a rod
- The motion of the particle may be described using one component
The constraint may be expressed in cartesian coordinates as
you can express the position of the object on the end of a rod as a function of just one generalized coordinate
- is the generalized coordinate
You could also express the position as a function of the deflection anglein cartesian coordinates
- is the generalized coordinate
- If you were to start using Polar coordinates right away such that
then the dependence of the function onwould not be obvious as this dependence is implicit to changes in the direction
The number of generalized coordinates becomes more obvious when you begin expressing the Potential and Kinetic Energy
Motion on Sphere example
Consider a particle constrained to move on a sphere of radius.
- There are 3 degrees of freedom for the 3-D problem
- The particle is constrained to the surface of the sphere
- The motion of the particle may be described using two components
The constraint expressed in terms of cartesian coordinates is
The above constraint equation can be used to reduce the degrees of freedom by using the constraint to eliminate one of the above components
or if you chose the angles and from spherical coordinates
Generalized Force and Momentum
- Generalized Force
- Generalized Momentum
As shown above, Hamilton's principle leads to a re-expression of Newton's second law through the Euler-Lagrange Equation in a differential form known as Lagrange's equations.
Below is the Euler-Lagrange equations expressed in terms of generalized coordinates.
If each coordinate used to describe a system can vary independently of the others, then the system is said to be holonomic.
A Non-holonomic system, has a lest one coordinate that depends on one of the others thereby reducing the number of degrees of freedom.
Example: Consider a sphere constrained to roll on a plane (ball on the floor). A spheres position on the plane can be specified using 2 coordinates and the orientation ( rotation) of a point on the sphere can be described by 3 coordinates. Thus, this system has 5 degrees of freedom.
The coordinates are not independent since as the sphere rolls without slipping then at least two coordinates must change making the system nonholonomic.
other examples: car, bicycle
Example: Lagrangian for object in 2-D moving in a conservative field
In cartesian coordinates
In cartesian coordinates the generalized coordinates are represented as the parameters x and y
- for the generalized coordinate labeled
- for the generalized coordinate labeled
In polar coordinates
In cartesian coordinates the generalized coordinates are represented as the parametersand
In polar coordinates
Examples using Lagrange's Equation
Returning to the problem of an object of massattached to a weightless rod of length that is constrained to rotate at the end of the rod where there is no mass.
The number of generalized coordinates = 2-1 = 1.
Selecting polar coordinates to describe this system allows a description using generalized coordinates of only one parameter,.
Let represent the distance of from the pulley and the distance of from the pulley.
The problem has 2 masses hat can move up and down giving us 2 degrees of freedom.
There is a constraint due to the masses being joined by a string of length l.
The number of generalized coordinate = 2-1 = 1.
One may define the potential arbitrarily such that the above constant is zero.
Block Sliding on a Wedge
A block slides down an incline of elevationthat is free to move on the table.
How long does it take the block to reach the bottom if it is released a distancefrom the top.
- distance of block from the top of the incline
- distance incline moves from its starting point.
- Kinetic energy of the block of mass m
- kinetic energy of the incline of mass M
The kinetic energy of the block is a combination of the velocity of the block down the incline and the velocity of the incline. As the block moves down the inline, the incline moves.
substituting q_2 into q_1
- If is 90 degrees then
The acceleration is constant so one may use constant acceleration equations to find the fall time
- If M is infinite then
The method of Lagrange multipliers is a means to incorporate the constraints of a system into the Euler-Lagrange equation.
consider the problem of a disk rolling down an incline plane without slipping.
If we assume the incline does not move then we have 2 degrees of freedom for the disk. Letrepresent the distance from the top of the incline and an angle of disk rotation.
The "no-slip" constraint means that there is a force of friction; which does no work by the way.
To roll without slipping requires that
The above constraint is holonomic since the constraint on the velocities can be integrated to give a relationship between the coordinates. Non-holonomic constraints do not have this property.
Constraints in Differential form
Using the above example we can generalize the constraint equation to take the form of a differential equation.
since the total derivative with respect to time is zero , the function is a constant in time
- A holonomic constraint (doesn't depend on other variables)
The advantage of having the constraint in differential form is that instead of incorporating their integrated form into the problem we add them to the Euler-Lagrange equation.
Incorporate the constraints into Euler-Lagrange Equations
Consider a system havingconstraints with generalized coordinate parameters.
For the case of time independent constraints, the-th constraint equation equation for the constraint function may be represented as
letbe an undetermined coefficient such that
The Euler-Lagrange equation is
we can add Lagrange multiplier to the above equation without changing it
but this is more than just equating terms that are equal to zero.
consider a system in cartesian coordinates such that
the above coordinates may be written in term of generalized coordinates or in other words they can be expressed as functions of q_i such that
- the same is try for y and z
the total derivative with respect to time is given by the chain rule as
- only the jth term survives the derivative of the sum
take derivative of both sides with respect to time
- here the total and partial derivatives are interchangeable
- Lagrange's equations of motion
Generalized forces of constraint
disk rolling down incline place using multipliers
Returning to the problem of a disk rolling without slipping down an incline place
consider a ladder of length L leaning against a frictionless wall and sitting on a frictionless floor.
Since everything is frictionless the ladder will begin to fall.
Find the equation of motion for the ladder.
There are 3 degrees of freedom; The theand position of the ladder and the angle, , the ladder makes with respect to the floor.
There are 2 constraint equations
But from the constraint equation f_1
But from the constraint equation f_2
substituting in forand
- = torque resulting from normal force of wall on ladder ( choose point of ladder's contact with floor as the point of rotation)
2-D Central Force
Consider a particle of massthat is constrained to move in 2-D due to a potential U(2).
Since this is a 2-D problem, lets use polar coordinates.
For thegeneralized coordinate parameter
for thegeneralized coordinate parameter
Particle confined on a cylinder
Consider a particle of massthat is experiencing a force and is constrained to move on a frictionless cylinder of radius .
Find the equation(s) to describe its motion.
Since the particle is constrained by a cylinder it seems worthwhile to use cylindrical coordinates to describe its motion.
to determine if the force is conservative.
The potential energy for this force may be defined according to the work integral
- simple harmonic motion in the z coordinate
- angular momentum is conserved
ball in a bowl
consider a ball of massand radius is constrained to roll without slipping spherical bowl of radius that is in a gravitational field.
The generalized coordinate functions for this problem are and where is the rotation angle of the ball and is the angle of inclination of the ball with respect to the cylinders central axis.
for thegeneralized coordinate function:
for the generalized coordinate function:
The constraint equation
- oscillation frequency
The double pendulum problem in 2-D has two objects (4 degrees of freedom) and two constraints.
The number of generalized coordinates = 4-2 = 2
Choosing the two anglesand
In cartesian coordinates one may write the position and velocity of the two objects as
- Object 1
- Object 2 is a little more complicated as its position depends on object 1
for the generalize coordinate paramter
- if I assume small oscillations
for thegeneralize coordinate paramter
- assuming small oscillations
Solving coupled ODEs
The double pendulum's motion for small angles is described by a set of coupled , non-linear, second order ordinary differential equations.
create a linear combination of the ODEs by adding them together using an arbitrary constant
Now require that the raio of the coefficients for the non-differential terms be equal to the ratio of the coefficients for the double differential terms
The differential equation for this linear combination may be written as
A normal coordinate is defined such that
Then the ODE looks like an oscillator
We have two possible solutions
"out of phase"
- if &
the "in phase" oscillation
the "out of phase" oscillation
- The frequency for the case when the two pendulums are oscillating in phase is lower than when they are out of phase.
Bead on a spinning hoop
A bead of massis threaded on a frictionless circular wire hoop or radius that is rotating at a constant velocity
The bead makes and angle with respect to the vertical.
Find equilibrium positions for the bead where it remains at a constantin terms of the centripetal acceleration ( )
The bead has two velocity components.
If the bead is moving up or down the circular hoop then
The bead is moving with the circular hoop as the hoop is rotated at a constant angular velocity
The Lagrangian for the bead is
For the bead to be in equilibrium the acceleration should be zero
if degrees then the bead is at the top or bottom of the hoop
another way is if
Then onlyand are equilibrium positions, is too small , the circular ring is rotating too slowly.
There are two cases depending on
- When and
- oscillating motion due to gravity pulling it down any movement away from the bottom of the hoop will result in a net force back towards the bottom of the hoop to
- oscillation frequency if bead is displaced a small amount from the bottom of the hoop
If the bead moves away fromthen you will have oscillation again but you won't return to as gravity is pulling the bead down and the normal force only cancels this at . This is an unstable critical point.
- When and
consider a small displacement from the above angle by and amount \epsilon
- :: is close to zero
- :: is close to zero
given equilibrium condition
- given equilibrium condition
Bifurcation: The splitting of a main body into two parts.
In the example above, as the hoop begins to rotate in the presence of a gravitational field we have one stable equilibrium at the bottom of the hoop. When the hoops rotating frequency reaches a critical valuethen the bead will be unstable at the bottom and seek to move up the hoop. There are two possible directions equilibrium points on either side of the hoop. The system is said to have bifricated at this point.