# Requirement of Holonomic Constraints for Deriving Lagrange Equations

• I
• deuteron
deuteron
TL;DR Summary
Why is it required for the constraints to be holonomic to derive the Euler-Lagrange equations of motion?
While deriving the Lagrange equations from d'Alembert's principle, we get from $$\displaystyle\sum_i(m\ddot x_i-F_i)\delta x_i=0\tag{1}$$ to $$\displaystyle\sum_k (\frac {\partial\mathcal L}{\partial\ q_k}-(\frac d {dt}\frac {\partial\mathcal L}{\partial\dot q_k}))\delta q_k=0\tag{2}$$

However, from the above step, we get to the below step only after assuming holonomic constraints:
$$(\frac {\partial\mathcal L}{\partial\ q_k}-(\frac d {dt}\frac {\partial\mathcal L}{\partial\dot q_k})=0.\tag{3}$$

Why is it that we have to assume holonomic constraints for that transition? My guess is that it has something to do with that if the constraints are not holonomic, then the virtual displacement are not always perpendicular to the trajectory of the body, but I can't see the mathematical connection between these.

PeroK
There is no restriction to holonomic constraints. You can also treat non-holonomic constraints. If done right, i.e., as a constraint on the "allowed" variations of the trajectories in configuration space, you get the same equations as from d'Alembert's principle. We have a lot of discussions on this in this forum. Just search for "vakonomic dynamics" ;-).

PeroK and deuteron

## What are holonomic constraints in the context of Lagrange equations?

Holonomic constraints are conditions that can be expressed as equations relating the coordinates and possibly time, in a form that can be integrated. These constraints reduce the number of independent coordinates needed to describe a system. For example, the constraint $$f(q_1, q_2, \ldots, q_n, t) = 0$$ is a holonomic constraint.

## Why are holonomic constraints required for deriving Lagrange equations?

Holonomic constraints are required because the Lagrangian formulation of mechanics relies on generalized coordinates that can describe the system's configuration space. These constraints allow us to reduce the degrees of freedom and simplify the equations of motion, making it possible to apply the principle of least action and derive the Lagrange equations.

## Can Lagrange equations be derived for systems with non-holonomic constraints?

Yes, Lagrange equations can be extended to handle non-holonomic constraints using methods such as Lagrange multipliers. However, the formulation becomes more complex because non-holonomic constraints cannot be integrated into a reduced set of coordinates and often involve inequalities or differential constraints.

## How do holonomic constraints simplify the problem of deriving Lagrange equations?

Holonomic constraints simplify the problem by reducing the number of independent variables or generalized coordinates needed to describe the system. This reduction allows us to write the kinetic and potential energy in terms of fewer variables, making the application of the Euler-Lagrange equation more straightforward.

## What is an example of a system with holonomic constraints?

An example of a system with holonomic constraints is a double pendulum. The lengths of the rods impose constraints on the positions of the masses, which can be expressed as equations relating the coordinates of the masses. These constraints reduce the number of independent coordinates from four (two for each mass) to two (the angles of the rods).

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