Constraint (classical mechanics)

As a constraint restricting the freedom of movement of a single- or multi-body system is known in analytical mechanics, in other words, a movement restriction. Thereby decreasing the number of degrees of freedom of a system. If too many constraints placed, it can happen that no physical solution exists.

The Lagrangian and Hamiltonian formulation of the classical mechanics or the D' Alembert's principle are particularly suitable to describe systems with constraints.

• 3.1 The pendulum
• 3.2 particles in spherical

Differentiation with respect to integrability

In the following a - particle system is always considered in three spatial dimensions. Without constraints were needed for the position vector of each particle 3 spatial coordinates, thus to describe the entire system as a whole. The coordinates are numbered consecutively:

Holonomic constraints

Holonomic constraints can be formulated as equations between the coordinates of the system ( holonomic constraints ):

The coordinates can be personalized with independent holonomic constraints on independent generalized coordinates, which must automatically satisfy the constraints that reduce:

Holonomic constraints are represented as the total differential of a function

And are therefore integrable. It is necessary for the integrability that the coefficient functions satisfy the following condition,

What needs to be added automatically at holonomic conditions ( twice continuously differentiable, see set of black).

Anholonome constraints

Nonholonomic / Anholonome constraints can not be formulated as equations between the coordinates. The generalized coordinates that appear in the anholonomen constraints, A., are not independently variable.

This example is to inequalities, such as restrictions on a particular region of space

Or differential, non- integrable constraints as equations between the speeds (eg for anholonome constraints )

Non- integrable means that the equation is not representable as a complete differential of a function as holonomic constraints. Thus, the coefficient functions do not satisfy the integrability condition here:

Differentiation with respect to time dependence

Further constraints are in terms of their time-dependence distinction as:

• Skleronom (rigid) if they do not explicitly depend on time.
• Rheonom (fluent), if they explicitly depend on time.

Scleronomic constraints result from applying the Lagrangian formalism usually result in a determination that the potential is not implicitly depend on time. If the potential is now also not explicitly time-dependent, so the forces are conservative and the energy is conserved. In this case, the Hamiltonian - the Legendre transform of the Lagrangian - equal to the total energy.

However holonomic - rheonome constraints can not directly conclude to a non- conservation of energy.

Examples

The pendulum

The cable of the pendulum to always have the same length, so it must satisfy the constraint due to the Pythagorean theorem. The deflection angle forms the generalized coordinate. This is an example of a holonomic constraint, and since they do not depend explicitly on time, for a scleronomic constraint.

Particles in spherical

It was locked a particle in a sphere. Which means mathematically that the distance of the particle to the center is always smaller than the radius R of the sphere must be, therefore applies. Since this is an inequality, the constraint condition is nonholonomic.