For a system with n degrees of freedom configuration space is n-dimensional. A point in configuration space is specified by giving n generalized coordinates. At any time, a system occupies a point in configuration space. At different times it may occupy different points. As time passes, the system point may move in configuration space.

Consider a systems of particles for which all forces, except the forces of
constraints, are derivable from a potential U(q) or U(q, dq/dt), and
for which the forces of constraints do no virtual work. For such a system
the equations of motion can be derived from different principles.

(i) Newton's laws

(ii) D'Alembert's principle

(iii) Hamilton's principle

Hamilton's principle deals with the motion of a system in configuration
space.

Assume that at time t_{1} the system point is P_{1} and at time
t_{2} the system point is P_{2}. There are many different
path from P_{1} to P_{2} that the system could have taken from P_{1}
to P_{2} in the time interval Δt = t_{2} - t_{1}.
Hamilton's principle states that the actual path taken by the system is the one
for which the line integral

I = ∫_{t1}^{t2}dt L(q, dq/dt, t)

has a stationary value (a minimum or a maximum). I is called the action and L is the Lagrangian function.

In general, if I is evaluated for some path of the system in configuration
space leading from (P_{1}, t_{1}) to (P_{2}, t_{2})
and then again for an infinitessimally different path, we have δI
≠ 0. But if I is evaluated for the actual path of the system and
then for an infinitessimally different path, we have

δI = δ∫_{t1}^{t2}dt L(q, dq/dt, t) = 0.

Using variational calculus, Hamilton's
principle leads to the following equation for systems for which all forces, except the
forces of constraints, are derivable from a potential and for which the forces
of constraints do no virtual work.

∫_{t1}^{t2}dt ∑_{k} [∂L/∂q_{k} - d/dt(∂L/∂(dq_{k}/dt))]
δq_{k} = 0.

If we have holonomic constraints and all the coordinates are independent,
then the δq_{k} are independent. and we have for each k

∂L/∂q_{k} - d/dt(∂L/∂(dq_{k}/dt)) =
0.

These are Lagrange's equations of motion. Lagrange's equations follow from Hamilton's principle.

We can always pick independent coordinates if we have holonomic constraints
of the form

f_{l}(q_{1},q_{2},q_{3},...,q_{n},t) =
0, l = 1, 2, 3, ... , m,

which reduce the number of the degree of freedom from n to k = n - m.

But does Hamilton's principle lead to the equations of motion if we do not pick independent coordinates, either because we do not have holonomic constraints, or because we do not use the constraints to eliminate the dependent coordinates?

Only in certain cases!

Assume we have n coordinates which are connected by m equations of constraints of the form

Σ_{k}a_{lk
}dq_{k
}+ a_{lt }dt = 0, l = 1, ..., m.

These equations connect the differentials of the coordinates q_{k}.
The coefficients a_{lk} and a_{lt} can be functions of the
coordinates and time.
Holonomic constraints
f(q_{1},q_{2},q_{3},...,q_{n},t) = 0 can always
be written in this form.

df =
Σ_{k}(∂f/∂q_{k})_{
}dq_{k
}+ (∂f/∂t)_{ }dt = 0.

But not all constraints that can be written in the form
Σ_{k}a_{lk
}dq_{k
}+ a_{lt }dt = 0 are a complete differential that can be
integrated.

If we now consider an infinitesimal virtual displacement from the actual path of the system, we have

Σ_{k}a_{lk
}δq_{k
}= 0,

since for virtual displacements δt = 0.

The variational process involved in Hamilton's principle concerns such virtual
displacements, since the time for each point on the path is held constant.

We may also write

λ_{l}Σ_{k}a_{lk
}δq_{k
}= 0,

with λ_{l} some undetermined quantity called an **
undetermined Lagrange multiplier**.

Combining this last equation with the equation we get from Hamilton's principle
we write

∫_{t1}^{t2}dt ∑_{k} [∂L/∂q_{k} - d/dt(∂L/∂(dq_{k}/dt))
+ λ_{l}Σ_{k}a_{lk}] δq_{k} = 0.

If we have n coordinates and m equations of constraints, then n - m
coordinates can be chosen independently.

If for the remaining m we choose

∂L/∂q_{k} - d/dt(∂L/∂(dq_{k}/dt)) + λ_{l}Σ_{k}a_{lk}
= 0, k = (n - m), .... n,

then the expression in parenthesis is zero for all k.

We can always make this choice, since we have m equations of constraints and m
undetermined multipliers λ.

In summary we have

d/dt(∂L/∂(dq_{k}/dt)) - ∂L/∂q_{k} = ∑_{l}λ_{l}a_{lk}, (n
equations),

and

Σ_{k}a_{lk
}dq_{k
}+ a_{lt }dt = 0, (m equations).

We have n + m equations for n + m unknowns, the n coordinates and the m λ's. We can solve these equations for the coordinates and the Lagrange multipliers.

We already know that if there are forces present that are not derivable from
a potential, then the equations of motions are

d/dt(∂L/∂(dq_{k}/dt)) -
∂L/∂q_{k} = Q_{k},

where the Q_{k} represents the generalized force not arising from a potential.

Therefore ∑_{l}λ_{l}a_{lk} must represent the generalized force of
constraint associated with the coordinate q_{k}.