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# Constrained Lagrangian Dynamics

Suppose that we have a dynamical system described by two generalized coordinates, and . Suppose, further, that and are not independent variables. In other words, and are connected via some constraint equation of the form
 (714)

where is some function of three variables. This type of constraint is called a holonomic. [A general holonomic constraint is of the form .] Let be the Lagrangian. How do we write the Lagrangian equations of motion of the system?

Well, according to Hamilton's principle,

 (715)

However, at any given instant in time, and are not independent. Indeed, Equation (714) yields
 (716)

at a fixed time. Eliminating from Equation (715), we obtain
 (717)

This equation must be satisfied for all possible perturbations , which implies that the term enclosed in curly brackets is zero. Hence, we obtain
 (718)

One obvious way in which we can solve this equation is to separately set both sides equal to the same function of time, which we shall denote . It follows that the Lagrangian equations of motion of the system can be written
 (719) (720)

In principle, the above two equations can be solved, together with the constraint equation (714), to give , , and the so-called Lagrange multiplier . Equation (719) can be rewritten
 (721)

Now, the generalized force conjugate to the generalized coordinate is [see Equation (599)]
 (722)

By analogy, it seems clear from Equation (721) that the generalized constraint force [i.e., the generalized force responsible for maintaining the constraint (714)] conjugate to takes the form
 (723)

with a similar expression for the generalized constraint force conjugate to .

Suppose, now, that we have a dynamical system described by generalized coordinates , for , which is subject to the holonomic constraint

 (724)

A simple extension of above analysis yields following the Lagrangian equations of motion of the system,
 (725)

for . As before,
 (726)

is the generalized constraint force conjugate to . Finally, the generalization to multiple holonomic constraints is straightforward.

Consider the following example. A cylinder of radius rolls without slipping down a plane inclined at an angle to the horizontal. Let represent the downward displacement of the center of mass of the cylinder parallel to the surface of the plane, and let represent the angle of rotation of the cylinder about its symmetry axis. The fact that the cylinder is rolling without slipping implies that and are interrelated via the well-known constraint

 (727)

The Lagrangian of the cylinder takes the form
 (728)

where is the cylinder's mass, its moment of inertia, and the acceleration due to gravity.

Note that and . Hence, Equation (725) yields the following Lagrangian equations of motion:

 (729) (730)

Equations (727), (729), and (730) can be solved to give
 (731) (732) (733)

The generalized constraint force conjugate to is
 (734)

This represents the frictional force acting parallel to the plane which impedes the downward acceleration of the cylinder, causing it to be less than the standard value . The generalized constraint force conjugate to is
 (735)

This represents the frictional torque acting on the cylinder which forces the cylinder to rotate in such a manner that the constraint (727) is always satisfied.

Consider a second example. A bead of mass slides without friction on a vertical circular hoop of radius . Let be the radial coordinate of the bead, and let be its angular coordinate, with the lowest point on the hoop corresponding to . Both coordinates are measured relative to the center of the hoop. Now, the bead is constrained to slide along the wire, which implies that

 (736)

Note that and . The Lagrangian of the system takes the form
 (737)

Hence, according to Equation (725), the Lagrangian equations of motion of the system are written
 (738) (739)

The second of these equations can be integrated (by multiplying by ), subject to the constraint (736), to give
 (740)

where is a constant. Let be the tangential velocity of the bead at the bottom of the hoop (i.e., at ). It follows that
 (741)

Equations (736), (738), and (741) can be combined to give
 (742)

Finally, the constraint force conjugate to is given by
 (743)

This represents the radial reaction exerted on the bead by the hoop. Of course, there is no constraint force conjugate to (since ) because the bead slides without friction.

Next: Hamilton's Equations Up: Hamiltonian Dynamics Previous: Hamilton's Principle
Richard Fitzpatrick 2011-03-31