What is configuration space and phase space?

Point in configuration space represents configuration of the system, i.e. positions of the constituent particles. Point in phase space represents state of the system, i.e. positions and velocities of the constituent particles together.

What is meant by configuration space in classical mechanics?

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.

What is the dimension of configuration space?

Normally, the configuration space is a continuous space (a “manifold”). The dimension of the configuration space is then called the number of degrees of freedom of the system. We shall see that a better characterization of what a system is doing at any given time is given by the phase space of the system.

Is configuration space a manifold?

In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints. The configuration space of a typical system has the structure of a manifold; for this reason it is also called the configuration manifold.

What is the configuration space of a double pendulum?

The double pendulum. The configuration space is thus the 2-torus T2 = S1 × S1.

What is configuration space in robotics?

The configuration space is a transformation from the physical space in which the robot is of finite-size into another space in which the robot is treated as a point. In other words, the configuration space is obtained by shrinking the robot to a point, while growing the obstacles by the size of the robot.

What is meant by phase space?

Definition of phase space : an ideal often multidimensional space of which the coordinate dimensions represent the variables required to specify the phase or state of a system or substance — see coordinate space.

What is the difference between state space and phase space?

A state space could be finite, consisting of just a few points. It could be finite-dimensional, consisting of an infinite number of points forming a smooth manifold, as usually the case in ordinary differential equations and mappings. Such a state space is often called a phase space.

What is a Lagrangian in physics?

Lagrangian function, also called Lagrangian, quantity that characterizes the state of a physical system. In mechanics, the Lagrangian function is just the kinetic energy (energy of motion) minus the potential energy (energy of position).

What is Lagrangian equation of motion?

Elegant and powerful methods have also been devised for solving dynamic problems with constraints. One of the best known is called Lagrange’s equations. The Lagrangian L is defined as L = T − V, where T is the kinetic energy and V the potential energy of the system in question.

What are constraints in classical mechanics?

In classical mechanics, a constraint on a system is a parameter that the system must obey. For example, a box sliding down a slope must remain on the slope. There are two different types of constraints: holonomic and non-holonomic.

What are configurations in math?

In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the same number of points.

Can trajectories in configuration space intersect?

Figure 3.1: A trajectory in a two-dimensional configuration space. It is possible for the trajectory to intersect itself, because the system can go back to the same position after a given interval of time.

What is spatial configuration in architecture?

Spatial configuration can be defined as a progressive process which connects the built environment and the human’s spatial experience and behaviour. Nevertheless, apart from this connection the configuration of space has also a cultural meaning which relates with the everyday life living patterns.

Who invented the double pendulum?

This two-mass system played a central role in the earliest historical development of dynamical equations of motion. Daniel Bernoulli, the son of Johann I Bernoulli, was the first to study the double pendulum, publishing a paper on the topic in 1733 in the proceedings of the Academy in St.

Why is a double pendulum chaotic?

A double pendulum executes simple harmonic motion (two normal modes) when displacements from equilibrium are small. However, when large displacements are imposed, the non-linear system becomes dramatically chaotic in its motion and demonstrates that deterministic systems are not necessarily predictable.

What is configuration space in path planning?

Path Planning Configuration space consists of free space, which contains all possible configurations (states) of the mobile system, and space occupied by obstacles.

What is C obstacle space?

Obstacles in c-space The insight is that a workspace obstacle is a region of space (a set of points) which corresponds to robot configurations that are not possible (or not allowed). An obstacle in c-space is just such a region in c-space: the set of points that are not legal configurations.

What is free space robotics?

The technology to substitute or subsidise the manned activities in space is called space robotics. Various applications of space robots are the inspection of a defective satellite, its repair, or the construction of a space station and supply goods to this station and its retrieval etc.

What do you mean by μ space and Γ space?

μ space and Γ space. Let us define μ space as phase space of one particle (atom or molecule) The Let us define μ – space as phase space of one particle (atom or molecule). The macrosystem phase space (Γ-space) is equal to the sum of μ – spaces.

Is phase space a vector space?

x is a 6N dimensional vector. Thus, the time evolution or trajectory of a system as specified by Hamilton’s equations of motion, can be expressed by giving the phase space vector, x as a function of time.

What is the volume of phase space?

In classical statistical mechanics, the statement (∆Ω)N = h3N is justified considering the product of an N independent quantum one-particle system or from the pure dimensional point of view: the Planck constant has dimensions of the two-dimensional phase-space volume x ×p, and each particle occupies a volume ∆Ω = h3.

What is the example of phase space?

So, for example, in a one dimensional system like a ball thrown directly upward or a spring, the phase space is two dimensional: position and momentum. For a trillion particles in a three dimensional box, the phase space has six trillion dimensions, three position and three momentum for each particle.

What is state space in quantum mechanics?

In physics, a state space is an abstract space in which different “positions” represent, not literal locations, but rather states of some physical system. This makes it a type of phase space. Specifically, in quantum mechanics a state space is a complex Hilbert space in which the possible instantaneous [ ? ]

What are state space equations?

In a state-space system, the internal state of the system is explicitly accounted for by an equation known as the state equation. The system output is given in terms of a combination of the current system state, and the current system input, through the output equation.

Do NOT follow this link or you will be banned from the site!