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 t₁ the system point is P₁ and at time t₂ the system point is P₂.  There are many different path from P₁ to P₂ that the system could have taken from P₁ to P₂ in the time interval Δt = t₂ - t₁.  Hamilton's principle states that the actual path taken by the system is the one for which the line integral

I = ∫_t1^t2dt 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 (P₁, t₁) to (P₂, t₂) 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^t2dt 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^t2dt ∑_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(q₁,q₂,q₃,...,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_ka_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(q₁,q₂,q₃,...,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_ka_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_ka_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_lΣ_ka_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^t2dt ∑_k [∂L/∂q_k - d/dt(∂L/∂(dq_k/dt)) + λ_lΣ_ka_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_lΣ_ka_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 = ∑_lλ_la_lk,  (n equations),

and

Σ_ka_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 ∑_lλ_la_lk must represent the generalized force of constraint associated with the coordinate q_k.