symplectic group (english) - meaning, definition, translation, pronunciation

Part of Speech

Noun

Phonetic Transcription

/sɪmˈplɛktɪk ɡruːp/

Meaning and Usage

The symplectic group, denoted as ( Sp(n) ), is a fundamental mathematical group that arises in the context of symplectic geometry and Hamiltonian mechanics. It is defined as the group of symplectic matrices, which are ( 2n \times 2n ) matrices that preserve a specific bilinear form known as the symplectic form.

The uses of the symplectic group are prevalent in various fields such as physics, particularly in classical mechanics and quantum mechanics. The study of the symplectic group is essential for understanding the transformation properties of Hamiltonian systems.

In terms of frequency of use, the term "symplectic group" is primarily encountered in written contexts, especially in academic papers, lecture notes, and textbooks related to advanced mathematics and theoretical physics.

Example Sentences

• The symplectic group plays a crucial role in the formulation of classical mechanics.
  (Група симплексних перетворень відіграє важливу роль у формулюванні класичної механіки.)
• Researchers often study the properties of the symplectic group to understand transformations in phase space.
  (Дослідники часто вивчають властивості симплексної групи для розуміння перетворень у фазовому просторі.)
• The configurations of the system can be related through the actions of the symplectic group.
  (Конфігурації системи можуть бути пов'язані через дії симплексної групи.)

Idiomatic Expressions

While "symplectic group" isn't commonly found in idiomatic expressions, it can be involved in phrases used within specialized academic contexts. Below are some phrases related to symplectic geometry that mention the symplectic group:

• The symplectic group acts transitively on the space of symplectic manifolds.
  (Симплексна група діє транситивно на просторі симплектичних маніфолдів.)
• Understanding the symplectic group is crucial for anyone studying Hamiltonian dynamics.
  (Розуміння симплексної групи є критично важливим для будь-кого, хто вивчає гамільтонівську динаміку.)
• The actions of the symplectic group can be visualized in phase space, leading to insights in classical mechanics.
  (Дії симплексної групи можна уявити в фазовому просторі, що призводить до нових знань у класичній механіці.)

Etymology

The term "symplectic" comes from the Greek word "symplektikos," meaning "to weave together." It reflects the group's role in intertwining geometrical structures in mathematics. The use of "group" relates to the algebraic structure entailing a set equipped with an operation that satisfies the group axioms.

Synonyms and Antonyms

Synonyms: (In the context of groups in mathematics)
- Symplectic matrix group

Antonyms:
- Inversely, there are no direct antonyms for the term "symplectic group," as it refers specifically to a certain algebraic structure within mathematics. However, in a broader mathematical context, you could consider groups without the symplectic property as indirect opposites (like orthogonal groups).

This structured information provides a comprehensive understanding of the term "symplectic group," its usage, importance, and relevance in mathematics.