Difference between revisions of "Forest UCM LEq"
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− | :<math>\vec r = x \hat i + y \hat j \pm \sqrt{R^2-x^2 - y^2} \hat k</math> | + | :<math>\vec r = x \hat i + y \hat j \pm \sqrt{R^2-x^2 - y^2} \hat k = \vec r ( x,y)</math> |
− | :<math> = </math> | + | |
+ | |||
+ | or if you chose the angles \theta and \phi from spherical coordinates | ||
+ | |||
+ | :<math> \vec r = R \cos \phi \sin \theta \hat i + R \sin \phi \sin \theta \hat j + R \cos \theta \hat k = \vec r (\phi, \theta) </math> | ||
Revision as of 13:47, 26 October 2014
Lagrange's Foramlism for Classical Mechanics
Hamilton's principle
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
or
here
- if I have conservative forces
- Newton's second law in an Inertial reference frame
thus
Lagrange's Equations in generalized coordinates
Generalized coordinates
are a set of coordinates that uniquely specify the instantaneous state of a dynamical system.The number of independent generalized coordinates
is given by subtracting the number of constraints from the number of degrees of freedom .Pedulum example
Consider the 2-D pendulum where an object of mass
is 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 angle
in cartesian coordinates- is the generalized coordinate
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
for Example
or if you chose the angles \theta and \phi from spherical coordinates
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.
The above is used to determine a stationary path followed by a particle obeying Newton's second law.
This path should be the same in all coordinate systems that are Inertial reference frames.