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Newton's Law of Motion01:20

Newton's Law of Motion

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When we observe objects around us, one question that comes to mind is why they move or stay still. The answer to this question can be explained using Newton's laws of motion. These laws describe the fundamental principles of motion and the effects of forces on objects.
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Newton's Second Law00:55

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Newton's second law is closely related to his first law of motion. It mathematically gives the cause-and-effect relationship between force and changes in motion. Newton's second law is quantitative and is used extensively to calculate what happens in situations involving a force. All external forces acting on a system add together to produce a net force Fnet. A larger net external force produces a larger acceleration. This acceleration is directly proportional to, and in the same...
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Newton's Law of Gravitational Attraction01:24

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Sir Isaac Newton established the universality of the law of gravitational attraction based on empirical evidence and inductive reasoning. He published his work in Philosophiae Naturalis Principia Mathematica ("the Principia") on July 5, 1687.
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Newton's First Law: Introduction01:17

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Motion draws our attention. Motion itself can be beautiful, causing us to marvel at the forces needed to create spectacular sights, such as that of a dolphin jumping out of the water, the flight of a bird, or the orbit of a satellite. The study of motion is kinematics, but kinematics only describes the way objects move—their velocity and acceleration. Dynamics considers the forces that affect the motion of moving objects and systems. Newton's laws of motion are the foundation of...
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Newton's First Law: Application01:12

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Experience suggests that an object at rest remains at rest if left alone, and that an object in motion tends to slow down and stop unless some effort is made to keep it moving. However, Newton's first law gives a deeper explanation of this observation. The study of Newton's laws is like recognizing patterns in nature from which further patterns can be discovered. The genius of Galileo, who first developed the idea for the first law of motion, and Newton, who clarified it, was to ask the...
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Newton's Third Law: Introduction00:58

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Whenever one body exerts a force on a second body, the first body experiences a force equal in magnitude and opposite in direction, to the force that it exerts. For instance, when a person pushes on a wall, the wall exerts an equal and opposite force towards the person. This brings us to Newton's third law of motion. Newton's third law represents a certain symmetry in nature: Forces always occur in pairs, and one body cannot exert a force on another without experiencing a force itself.
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Methods for Presenting Real-world Objects Under Controlled Laboratory Conditions
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The Set-Based Hypervolume Newton Method for Bi-Objective Optimization.

Victor Adrian Sosa Hernandez, Oliver Schutze, Hao Wang

    IEEE Transactions on Cybernetics
    |January 1, 2019
    PubMed
    Summary
    This summary is machine-generated.

    This study introduces the hypervolume Newton method (HNM) for approximating the Pareto front of bi-objective optimization problems. A hybrid approach combining HNM with evolutionary strategies offers a fast and reliable solution for hypervolume maximization.

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    Area of Science:

    • Optimization
    • Numerical Analysis
    • Computational Science

    Background:

    • Approximating the Pareto front (PF) is crucial for bi-objective optimization problems (BOPs).
    • The hypervolume indicator is a standard metric for evaluating the quality of PF approximations.
    • Existing methods may lack efficiency or reliability in hypervolume maximization.

    Purpose of the Study:

    • To develop a set-based Newton method for computing finite-size approximations of the Pareto front.
    • To introduce the hypervolume Newton method (HNM) for efficient hypervolume maximization.
    • To propose a hybrid algorithm combining HNM with evolutionary strategies for improved performance.

    Main Methods:

    • Analytical derivation of the Hessian matrix for the hypervolume indicator.
    • Development of the hypervolume Newton method (HNM) for unconstrained and inequality-constrained BOPs.
    • Hybridization of HNM with evolutionary strategies to enhance convergence and reliability.

    Main Results:

    • The HNM demonstrates potential for quadratic convergence to optimal solutions.
    • The hybrid HNM algorithm provides a fast and reliable approach for hypervolume maximization.
    • Performance of HNM and the hybrid method is validated on benchmark problems.

    Conclusions:

    • The proposed hypervolume Newton method offers an efficient approach to Pareto front approximation.
    • Hybridization with evolutionary strategies enhances the robustness and speed of the method.
    • The developed algorithms represent a significant advancement in hypervolume maximization for BOPs.