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Pappus and Guldinus's theorems are powerful mathematical principles that are used for finding the surface area and volume of composite shapes. For example, consider a cylindrical storage tank with a conical top. Finding the surface area or volume can be challenging for such complex shapes. These theorems are particularly useful in calculating the volume and surface area of such systems. Here, the cylindrical storage tank with a conical top can be broken down into two simple shapes: a...
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Once the fields have been calculated using Maxwell's four equations, the Lorentz force equation gives the force that the fields exert on a charged particle moving with a certain velocity. The Lorentz force equation combines the force of the electric field and of the magnetic field on the moving charge. Maxwell's equations and the Lorentz force law together encompass all the laws of electricity and magnetism. The symmetry that Maxwell introduced into his mathematical framework may not be...
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The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write...
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The two theorems developed by Pappus and Guldinus are widely used in mathematics, engineering, and physics to find the surface area and volume of any body of revolution. This is done by revolving a plane curve around an axis that does not intersect the curve to find its surface area or revolving a plane area around a non-intersecting axis to calculate its volume.
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Setting Limits on Supersymmetry Using Simplified Models
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The Symplectic Camel and Poincaré Superrecurrence: Open Problems.

Maurice A de Gosson1

  • 1Faculty of Mathematics, NuHAG, University of Vienna, 1090 Vienna, Austria.

Entropy (Basel, Switzerland)
|December 3, 2020
PubMed
Summary

Poincaré

Area of Science:

  • Mathematical Physics
  • Dynamical Systems
  • Symplectic Topology

Background:

  • Poincaré's Recurrence Theorem states isolated Hamiltonian systems in bounded spaces return to initial states.
  • Recurrence properties are fundamental to understanding the long-term behavior of classical and quantum systems.

Purpose of the Study:

  • To explore Poincaré's Recurrence Theorem and related properties using recent advances in symplectic topology.
  • To bridge the gap between classical mechanics and quantum mechanics through the lens of recurrence.
  • To highlight the relevance of symplectic topology to the physics community.

Main Methods:

  • Review and synthesis of recent developments in symplectic topology.
  • Application of topological concepts to analyze recurrence in Hamiltonian systems.
Keywords:
HamiltonianPoincaré recurrencequantum mechanicssymplectic camel

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  • Conceptual exploration of the connection between classical recurrence and quantum phenomena.

    • Main Results:

    • Symplectic topology offers novel perspectives on recurrence properties in dynamical systems.
    • These topological insights can illuminate the transition from classical to quantum mechanics.
    • The study identifies a connection between recurrence and emergent quantum phenomena.

    Conclusions:

    • Recent advances in symplectic topology provide powerful tools for studying recurrence in physics.
    • Understanding recurrence is crucial for developing theories of emergent quantum mechanics.
    • Interdisciplinary approaches combining topology and physics can yield significant insights.