Advanced Classical Mechanics/Constraints and Lagrange's Equations

There is more to classical mechanics than . Often the motion of a system is constrained in some way.

What are constraints?Edit

  • The particles could be restricted to travel along a curve or surface. Specifically one could have some function of the coordinates of each particle and time vanish. These restrictions are either kinematical or geometrical in nature.

 

This is called a holonomic constraint. For example we could have

 

which expresses that the distances between two particles that make up a rigid body are fixed.

  • There are non-holonomic constraints. For example, one could have

  for a particle travelling outside the surface of a sphere or constraints that depend on velocities as well,

 

A familar example of the latter is a ball rolling on a surface.

We will be dealing exclusively with holonomic constraints in this course.

What are the consequences?Edit

  • The coordinates are no longer independent. They are related through the equations of constraint.
  • The force of constraint are not given so they must be determined from the solution (if you actually want them at all).
  • If the constraints are holonomic, the equations of constraint can be used to eliminate some of the coordinates to get a set of generalized independent coordinates.

These generalized coordinates usually will not fall into pairs or triples that transform as vectors, e.g. the natural coordinates for motion restricted to a sphere are spherical coordinates ( ).

D'Alembert's PrincipleEdit

D'Alembert's principle relies on the concept of virtual displacements. The idea is that you can imagine freezing the system in time and jiggling each of the particles in a way consistent with the various constraints at that particular time and determine the work (virtual work) needed to perform these virtual displacements.

We will denote the virtual displacement of a particle as  . The virtual displacement must be consistent with the constraints.

For each particle we have

 

and summing over the particles we have

 

Let's divide the force on each particle into applied forces and constraints

 

so we have

 

If we assume that the forces of constraint do no virtual work, then the last term vanishes. This is a reasonable assumption because the force of constraint to restrict a particle to a surface is normal to that surface but a displacement consistent with the constraints is tangent to the surface so the dot product will vanish and the force of constraint performs no virtual work. If the constraints are a function of time, the forces of constraint can perform work on the particles but the whole idea of D'Alembert's principle is that we have frozen time.

This leaves us with

 

D'Alembert's principle. The forces of constraints are gone! However, the coordinates are not independent, so the equation is only a statement about the sum of the various forces, momenta and virtual displacements.

Generalized CoordinatesEdit

We can try to find a set of independent coordinates given the constraints. Let's write

 

and so

 

and a virtual displacement is

 

Notice that there is no   in the virtual displacement because time is held fixed. Now let's look at the first term in D'Alembert's equation

 

where   is called a generalized force.

The second term takes a bit more work. We have

 

Let's focus on a particular one of the generalized coordinates j. We have

 

                               

We can use the fact that

 

from the defintion of   to get

                               

                               

Notice that the quantity in the innermost parenthesis is just the total kinetic energy of the system   so we have

 

Lagrange's EquationsEdit

If the constraints are holonomic, we can pick the   to be independent so the various   are completely arbitrary and the quantity in braces must vanish to yield

 

These expressions are sometimes called Lagrange's equations, but the term Lagrange's equations is often reserved for the case of a conservative force. In this case we have

  so

 

In this case we can rearrange the equation to give

 

Furthermore, if the forces do not depend explicitly on the velocities we can define, the Lagrangian to be  

and write Lagrange's equations in their traditional form

 

Conserved QuantitiesEdit

For each coordinate we can define a conjugate momentum to be

 

This is called the generalized momentum conjugate to the coordinate  . Let's look at Lagrange's equations with this definition

 

so if the Lagrangian does not depend on a particular coordinate  , then the momentum conjugate to the coordinate does not change with time; it is conserved. This conserved momentum is called a first integral. The coordinate that doesn't affect the Lagrangian is called a cyclic coordinate.

In general the Lagrangian will depend on the coordinates, velocities and time; what happens if   vanishes? Is there a conserved quantity similar to the momenta?

Let calculate the total derivative of the Lagrangian with respect to time,

 

Now let's use the definition of the momenta and the Lagrange's equation to simplify things a bit,

 

and rearranging

 

So if

 

then the Hamiltonian,

 

is conserved.

Lagrangians with Non-Conservative ForcesEdit

From the analysis so far it would appear that one can only construct a Lagrangian when the forces that act on a particle are conservative (they can be derived from a potential  ). It is indeed the case that truly dissipative forces such as friction cannot be directly included in a Lagrangian formulation, but forces that can be written in the form   may be included in the Lagrangian. Although this seems very restrictive, an important force of this class is the magnetic force on a charged particle.

 

In electrostatics the electric field is simply the gradient of the electrostatic potential, but for more general fields we have

 

where   is the vector potential. The magnetic field may be written as

 

Let's consider the function

 

and calculate

 

The total time derivative of the vector potential generates several terms (due to the chain rule) to yield

 

Notice that the terms proportional to   cancel leaving

 

The first term is simply the charge of the particle times the x-component of the electric field. The second term is the charge of the particles times the x-component of  , so the following Lagrangian will yield the equations of motion

 

If the vector potential and the scalar potential do not depend on time, then   and

 

is conserved. The force is not conservative but the system is.