Lagrange multipliers: holonomic constraints
In Lagrangian mechanics, it is possible to account for systems where generalized coordinates \mathbf{q} = (q_1, ..., q_N) are subject to holonomic constraints. These constraints take the form:
g_k(\mathbf{q}, t) = 0, \quad k = 1, ..., m
meaning not all coordinates are independent during the system’s motion. The method of Lagrange multipliers lets me enforce these constraints directly at the level of the equations of motion, without the need to eliminate dependent variables.
I start by constructing the standard Lagrangian, \mathcal{L}(\mathbf{q}, \dot{\mathbf{q}}, t) = T - V, where T is the kinetic and V is the potential energy. To treat the constraints, I introduce one Lagrange multiplier \lambda_k(t) for each constraint and form the augmented Lagrangian:
\mathcal{L}^\prime(\mathbf{q}, \dot{\mathbf{q}}, \boldsymbol\lambda, t) = \mathcal{L}(\mathbf{q}, \dot{\mathbf{q}}, t) - \sum_{k=1}^{m} \lambda_k(t)\,g_k(\mathbf{q}, t)
I then apply the Euler-Lagrange equation to each q_i as if all the coordinates were independent. Because holonomic constraints do not depend on velocities, the equations reduce to:
\frac{\mathrm d}{\mathrm dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{q}_i} \right) - \frac{\partial \mathcal{L}}{\partial q_i} = -\sum_{k=1}^{m} \lambda_k \frac{\partial g_k}{\partial q_i}
for i = 1,...,N. The right-hand side represents the generalized constraint forces, highlighted explicitly by the presence of the multipliers and derivatives of the constraint functions.
For each multiplier, I also require:
g_k(\mathbf{q}, t) = 0
which simply maintains the holonomic constraint at all times.
Solving the full system gives the N coordinate trajectories and the m functions \lambda_k(t). The multipliers have a direct interpretation as measures of the force needed to maintain each constraint along the motion.
This procedure forms a consistent and efficient approach to constrained motion in Lagrangian mechanics.
For more insights into this topic, you can find the details here.