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Related Concept Videos

Energy Diagrams - II01:10

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Energy diagrams are important to understand the dynamics of a system. The topology of an energy diagram helps illustrate the equilibrium points of the system.
The point in the energy diagram at which the system’s potential energy is the lowest is known as the local minima. The system tends to stay in this position indefinitely unless acted upon by a net force. The slope of the potential energy diagram at the local minima is zero, indicating that zero net force is acting on the system. The...
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The energy stored by a structure and location of matter in space is called potential energy. For instance, raising a kettlebell changes its spatial location and increases its potential energy. Similarly, a stretched rubber band contains potential energy which, under certain conditions, can be converted into other forms of energy, such as kinetic energy.
Chemical bonds that form attractive forces between atoms also contain potential energy, called chemical energy. When a chemical reaction...
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A conservative force, such as a gravitational or elastic force, gives the body the capacity to do work. This capacity, measured as the potential energy, depends on the body's location or “position” relative to a fixed reference position or datum. The gravitational potential energy is considered zero at the reference point. Suppose a body is located at some vertical distance above a fixed horizontal reference or datum. In that case, the weight of the body has positive gravitational potential...
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Potential energy or potential function plays an essential role in determining the stability of a mechanical system. If a system is subjected to both gravitational and elastic forces, the potential function of the system can be expressed as the algebraic sum of gravitational and elastic potential energy. If the system is in equilibrium and is displaced by a small amount, then the work done on the system equals the negative of the change in the system's potential energy from the initial to the...
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The dynamics of a mechanical system can be easily understood by interpreting a potential energy diagram. Since energy is a scalar quantity, the interpretation of the dynamics of the system becomes even simpler.
Take the example of a skater on a parabolic ramp. The potential energy at different points along the ramp will be proportional to the height of the ramp, which varies quadratically with the horizontal position on the ramp. As the skater moves down the ramp from the highest position,...
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Potential energy is also known as energy at rest or stored energy. Common types of potential energy include the gravitational potential energy stored in an apple hanging from a tree, the electrical potential energy stored in an object due to the attraction or repulsion of electric charges, and the chemical potential energy stored in the bonds between atoms and molecules. Additionally, the nuclear energy stored in an atomic nucleus and the elastic energy stored in a stretched spring due to its...
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Certification and the potential energy landscape.

Dhagash Mehta1, Jonathan D Hauenstein1, David J Wales2

  • 1Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695, USA.

The Journal of Chemical Physics
|June 16, 2014
PubMed
Summary
This summary is machine-generated.

This study introduces certification methods to guarantee numerical solutions for nonlinear equations. Smale

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Area of Science:

  • Computational Mathematics
  • Numerical Analysis
  • Theoretical Chemistry

Background:

  • Standard nonlinear equation solvers may yield approximations that do not converge to true solutions.
  • Numerical approximations can fall into linear or chaotic convergence basins, leading to misleading results.
  • Certification is crucial to mathematically prove that a numerical approximation is an actual solution.

Purpose of the Study:

  • To present a method for certifying numerical stationary points of potential energy landscapes.
  • To provide a mathematical proof for the quadratic convergence of numerical approximations.
  • To ensure the reliability of numerical solutions in complex optimization problems.

Main Methods:

  • Application of Smale's α-theory for certification.
  • Mathematical analysis of convergence properties.
  • Verification of stationary points in potential energy landscapes.

Main Results:

  • Demonstrated the efficacy of Smale's α-theory in certifying numerical solutions.
  • Provided a rigorous mathematical framework for validating approximations.
  • Ensured that certified stationary points correspond to actual solutions, irrespective of numerical precision.

Conclusions:

  • Smale's α-theory offers a robust approach to certifying numerical stationary points.
  • This method enhances the reliability of computational results in scientific research.
  • Certification guarantees that numerical approximations are true solutions, avoiding misleading outcomes.