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The set of coordinates that define the position of a reference point and the orientation of a coordinate frame attached to a rigid body in three-dimensional space form its configuration space, often denoted () where represents the coordinates of the origin of the frame attached to the body, and () represents the rotation matrices that define the orientation of this frame relative to a ground ...
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system, whose origin, orientation, and scale have been specified in physical space. It is based on a set of reference points , defined as geometric points whose position is identified both mathematically (with numerical coordinate values) and ...
Thermodynamic work is one of the principal kinds of process by which a thermodynamic system can interact with and transfer energy to its surroundings. This results in externally measurable macroscopic forces on the system's surroundings, which can cause mechanical work, to lift a weight, for example, [1] or cause changes in electromagnetic, [2] [3] [4] or gravitational [5] variables.
The ancient Greek understanding of physics was limited to the statics of simple machines (the balance of forces), and did not include dynamics or the concept of work. During the Renaissance the dynamics of the Mechanical Powers, as the simple machines were called, began to be studied from the standpoint of how far they could lift a load, in addition to the force they could apply, leading ...
In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane. For every possible state of the system or allowed combination of values of the system's parameters, a point is ...
In the new coordinate system, the origin is a fixed point of the map and the solutions are of the linear system A n x 0. The solutions for the map are no longer curves, but points that hop in the phase space. The orbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map.
A map from this sphere to a unit sphere of dimension n − 1 can be constructed by dividing each vector on this sphere by its length to form a unit length vector, which is a point on the unit sphere S n−1. This defines a continuous map from S to S n−1. The index of the vector field at the point is the degree of this map. It can be shown ...
The PES concept finds application in fields such as physics, chemistry and biochemistry, especially in the theoretical sub-branches of these subjects. It can be used to theoretically explore properties of structures composed of atoms, for example, finding the minimum energy shape of a molecule or computing the rates of a chemical reaction .