- #1

deuteron

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- TL;DR Summary
- why are the gradients of the holonomic constraints perpendicular to the constraint forces

My question is about the general relationship between the constraint functions and the constraint forces, but I found it easier to explain my problem over the example of a double pendulum:

Consider a double pendulum with the generalized coordinates ##q=\{l_1,\theta_1,l_2,\theta_2\}##,:

The set of constraint functions is:

$$f=\begin{pmatrix} l_1-\text{const.}_1\\ l_2-\text{const.}_2\end{pmatrix}=0$$

Since ##f=0## describes the level curve ##N_0(f)##, it describes a submanifold in the configuration space, the generalized coordinates of phase space

Since this is a level curve, ##\nabla f## is perpendicular to the manifold embedded in the configuration space

However, the constraint *forces* act on the physical plane of motion, which is a submanifold in ##3d## space

Therefore, I don't understand how we can say that ##\nabla f\| F_\text{constraint}## in the Lagrangian mechanics, since they act on manifolds embedded in different spaces

Consider a double pendulum with the generalized coordinates ##q=\{l_1,\theta_1,l_2,\theta_2\}##,:

The set of constraint functions is:

$$f=\begin{pmatrix} l_1-\text{const.}_1\\ l_2-\text{const.}_2\end{pmatrix}=0$$

Since ##f=0## describes the level curve ##N_0(f)##, it describes a submanifold in the configuration space, the generalized coordinates of phase space

Since this is a level curve, ##\nabla f## is perpendicular to the manifold embedded in the configuration space

However, the constraint *forces* act on the physical plane of motion, which is a submanifold in ##3d## space

Therefore, I don't understand how we can say that ##\nabla f\| F_\text{constraint}## in the Lagrangian mechanics, since they act on manifolds embedded in different spaces