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Concept
Holonomic constraints
Summary
In classical mechanics, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following form:
where are n generalized coordinates that describe the system (in unconstrained configuration space). For example, the motion of a particle constrained to lie on the surface of a sphere is subject to a holonomic constraint, but if the particle is able to fall off the sphere under the influence of gravity, the constraint becomes non-holonomic. For the first case, the holonomic constraint may be given by the equation
where is the distance from the centre of a sphere of radius , whereas the second non-holonomic case may be given by
Velocity-dependent constraints (also called semi-holonomic constraints) such as
are not usually holonomic.
In classical mechanics a system may be defined as holonomic if all constraints of the system are holonomic. For a constraint to be holonomic it must be expressible as a function:
i.e. a holonomic constraint depends only on the coordinates and maybe time . It does not depend on the velocities or any higher-order derivative with respect to t. A constraint that cannot be expressed in the form shown above is a nonholonomic constraint.
As described above, a holonomic system is (simply speaking) a system in which one can deduce the state of a system by knowing only the change of positions of the components of the system over time, but not needing to know the velocity or in what order the components moved relative to each other. In contrast, a nonholonomic system is often a system where the velocities of the components over time must be known to be able to determine the change of state of the system, or a system where a moving part is not able to be bound to a constraint surface, real or imaginary. Examples of holonomic systems are gantry cranes, pendulums, and robotic arms. Examples of nonholonomic systems are Segways, unicycles, and automobiles.
The configuration space lists the displacement of the components of the system, one for each degree of freedom.
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Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his 1788 work, Mécanique analytique. Lagrangian mechanics describes a mechanical system as a pair consisting of a configuration space and a smooth function within that space called a Lagrangian. For many systems, where and are the kinetic and potential energy of the system, respectively.
Classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical mechanics, if the present state is known, it is possible to predict how it will move in the future (determinism), and how it has moved in the past (reversibility). The "classical" in "classical mechanics" does not refer classical antiquity, as it might in, say, classical architecture.
Configuration space (physics)
In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the space defined by these coordinates is called the configuration space of the physical system. It is often the case that these parameters satisfy mathematical constraints, such that the set of actual configurations of the system is a manifold in the space of generalized coordinates. This manifold is called the configuration manifold of the system. Notice that this is a notion of "unrestricted" configuration space, i.