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Related Concept Videos

Gauss's Law: Problem-Solving01:10

Gauss's Law: Problem-Solving

Gauss's law helps determine electric fields even though the law is not directly about electric fields but electric flux. In situations with certain symmetries (spherical, cylindrical, or planar) in the charge distribution, the electric field can be deduced based on the knowledge of the electric flux. In these systems, we can find a Gaussian surface S over which the electric field has a constant magnitude. Furthermore, suppose the electric field is parallel (or antiparallel) to the area vector...
Gauss's Law01:07

Gauss's Law

If a closed surface does not have any charge inside where an electric field line can terminate, then the electric field line entering the surface at one point must necessarily exit at some other point of the surface. Therefore, if a closed surface does not have any charges inside the enclosed volume, then the electric flux through the surface is zero. What happens to the electric flux if there are some charges inside the enclosed volume? Gauss's law gives a quantitative answer to this question.
Equations of Motion: Rectangular Coordinates and Cylindrical Coordinates01:21

Equations of Motion: Rectangular Coordinates and Cylindrical Coordinates

Understanding the motion of particles is a fundamental aspect of classical mechanics, and the choice of the coordinate system plays a pivotal role in unraveling the complexities of their dynamics.
When a particle moves relative to an inertial frame, the equations of motion can be expressed using rectangular components. If the motion is confined to the x-y plane, the equations having the x and y coordinates only can be used to simplify the mathematical representation.
However, when particles...
Gauss's Law: Cylindrical Symmetry01:20

Gauss's Law: Cylindrical Symmetry

A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...
Equations of Equilibrium in Three Dimensions01:30

Equations of Equilibrium in Three Dimensions

When analyzing structures or systems at rest, it is necessary to ensure they are in equilibrium. This is where the vector and scalar equations of equilibrium come into play. These equations are crucial in ensuring a structure is stable and will not collapse or fall apart. The vector and scalar equations of equilibrium provide a framework for analyzing the forces acting on a body.
According to the vector equations of equilibrium, the vector sum of all the external forces acting on a body must...
Euler Equations of Motion01:19

Euler Equations of Motion

Imagine a rigid body that is rotating at an angular velocity of ω within an inertial frame of reference. Along with this, picture a second rotating frame that is attached to the body itself. This frame moves along with the body and possesses an angular velocity of Ω. The total moment about the center of mass is calculated by adding the rate of change of angular momentum about the center of mass in relation to the rotating frame and the cross-product of the body's angular velocity and its...

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Related Experiment Video

Updated: Jun 19, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Nonholonomic double-bracket equations and the Gauss thermostat.

Alberto G Rojo1, Anthony M Bloch

  • 1Department of Physics, Oakland University, Rochester, Michigan 48309, USA. rojo@oakland.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 2, 2009
PubMed
Summary

This study introduces a new class of nonlinear equations derived from a constant kinetic-energy constraint on oscillators. These equations generalize double-bracket equations and offer insights into nonequilibrium molecular dynamics.

Related Experiment Videos

Last Updated: Jun 19, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Area of Science:

  • Nonlinear dynamics
  • Statistical mechanics
  • Mathematical physics

Background:

  • Nonlinear nonholonomic constraints are crucial in modeling complex systems.
  • Nonequilibrium molecular dynamics requires advanced theoretical frameworks.
  • Double-bracket equations are significant in integrable systems and gradient flows.

Purpose of the Study:

  • To investigate equations arising from a constant kinetic-energy constraint on 1D oscillators.
  • To explore the connection between these equations and generalized double-bracket equations.
  • To analyze the dynamics and solutions under specific potential conditions.

Main Methods:

  • Imposing a nonlinear nonholonomic constraint (constant kinetic energy).
  • Applying Gauss's law of least constraint for dynamics.
  • Analyzing the resulting equations for specific external potentials (e.g., harmonic).

Main Results:

  • The derived equations generalize the double-bracket equations.
  • Specific external potentials lead to a symmetric bracket description.
  • Periodic solutions were obtained for harmonic potentials.

Conclusions:

  • The constant kinetic-energy constraint provides a novel route to generalized integrable systems.
  • The findings are relevant for nonequilibrium molecular dynamics simulations.
  • Periodic solutions highlight potential for stable dynamics in constrained oscillator systems.