Physics+ Lagrangian Constraints: UNIZOR.COM - Physics+ 4 All - Lagrangian

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Constraints

Consider a mechanical system moving in n-dimensional configuration space and described by n time-dependent coordinates:
s1,...,sn.
Assume, there are m constraints that affect the motion of this system
f(1)(s1,...,sn)=0,
...
f(m)(s1,...,sn)=0.

Then under certain conditions (see below) the movement of this system can be described by n−m generalized parameters, which means that the system has n−m degrees of freedom.
These conditions include:
(a) constraints must be holonomic (expressed as equations that contain functions of only coordinates and time, not the velocities or other non-positional parameters);
(b) constraints must be independent (not derived from one another) which for holonomic constraints would follow from the requirement of linear independency of their gradients.

It should be noted that each individual constraint, provided its gradient is nonzero, defines a surface (a smooth manifold of dimension n−1) in a configuration space.
A trajectory of a system constrained by m holonomic constraints with linearly independent gradients belongs to an intersection of all such surfaces (which is a smooth manifold of dimension n−m) defined by all m constraints.
The system's velocity at any point on its trajectory must be tangential to this intersection of surfaces and, therefore, perpendicular to each constraint's gradient.
With m linearly independent gradients that together span an m-dimensional space, the tangent space where all trajectories belong to span orthogonal to it (n−m)-dimensional manifold, and generalized coordinates are just coordinates in this constrained manifold.