Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Integration by Parts: Definite Integrals01:23

Integration by Parts: Definite Integrals

Definite integrals involving the product of two functions over a fixed interval can be evaluated using integration by parts. This method rewrites the integral as the difference of a product evaluated at the endpoints and a remaining definite integral that is often simpler to compute.A representative example is the definite integral of the inverse tangent function. Since there is no direct integration formula for arctan ⁡x, the integrand is rewritten as a product of arctan⁡ x and the constant...
Fundamental Theorem of Calculus I01:23

Fundamental Theorem of Calculus I

Solving problems involving definite integrals requires a systematic approach that ensures clarity and efficiency. The first step is understanding the problem by identifying the calculated quantity, whether it involves accumulation, area, or a physical concept like force or probability. It is essential to recognize given conditions, such as the range of integration and any constraints that may affect the solution. Before computing, key properties of definite integrals should be analyzed to...
Separable Differential Equations01:20

Separable Differential Equations

A separable differential equation is a type of first-order differential equation where the derivative dy/dx can be expressed as a product of two functions: one that depends only on x and another that depends only on y. This allows for the rearrangement of the equation so that all terms involving y are on one side, and all terms involving x are on the other. This process, known as the separation of variables, simplifies the process of solving the equation by enabling the integration of both...
Rationalizing Substitutions01:29

Rationalizing Substitutions

Integrals involving non-rational functions are often difficult to evaluate using standard techniques, especially when radicals appear in the integrand. Rationalizing substitution provides a systematic method for simplifying such integrals by converting them into rational forms that are easier to handle.Consider a rod whose linear mass density depends on a constant linear density, a characteristic length, and the distance from the left end of the rod. Determining the total mass requires...
Improper Integrals: Discontinuous Integrands01:28

Improper Integrals: Discontinuous Integrands

Evaluating Areas Under Curves with DiscontinuitiesA definite integral is considered improper when the integrand is discontinuous at one of the limits of integration. This occurs when the function is undefined or becomes infinite at an endpoint, making the corresponding region under the curve unbounded. Such behavior is commonly associated with vertical asymptotes at the boundary of the interval. To properly define and evaluate these integrals, a limiting process is used to determine whether a...
Substitution Rule Applied to Definite Integrals01:24

Substitution Rule Applied to Definite Integrals

When evaluating a definite integral whose integrand matches the structure of a composite function, the substitution method provides an efficient way to simplify the calculation. This method is based on reversing the chain rule from differentiation, allowing a complicated expression to be rewritten in a simpler form. When the integrand contains an inner function and its derivative, substitution naturally reduces the complexity of the problem.The core idea of substitution for definite integrals...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Group Structure as a Foundation for Entropies.

Entropy (Basel, Switzerland)·2024
Same author

Information Geometry, Complexity Measures and Data Analysis.

Entropy (Basel, Switzerland)·2022
Same author

Permutation group entropy: A new route to complexity for real-valued processes.

Chaos (Woodbury, N.Y.)·2022
Same author

A generalized permutation entropy for noisy dynamics and random processes.

Chaos (Woodbury, N.Y.)·2021
Same author

Multivariate group entropies, super-exponentially growing complex systems, and functional equations.

Chaos (Woodbury, N.Y.)·2020
Same author

Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory.

Entropy (Basel, Switzerland)·2020

Related Experiment Video

Updated: May 21, 2026

Setting Limits on Supersymmetry Using Simplified Models
07:46

Setting Limits on Supersymmetry Using Simplified Models

Published on: November 15, 2013

Generalized Lenard chains, separation of variables, and superintegrability.

Piergiulio Tempesta1, Giorgio Tondo

  • 1Departamento de Física Teórica II, Facultad de Físicas, Universidad Complutense, 28040 Madrid, Spain. p.tempesta@fis.ucm.es

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 12, 2012
PubMed
Summary

Generalized Lenard chains provide a framework for multiseparable and superintegrable systems in bi-Hamiltonian geometry. Their existence on a four-dimensional manifold guarantees separation of variables, with applications to known systems.

More Related Videos

Spatial Separation of Molecular Conformers and Clusters
10:37

Spatial Separation of Molecular Conformers and Clusters

Published on: January 9, 2014

Related Experiment Videos

Last Updated: May 21, 2026

Setting Limits on Supersymmetry Using Simplified Models
07:46

Setting Limits on Supersymmetry Using Simplified Models

Published on: November 15, 2013

Spatial Separation of Molecular Conformers and Clusters
10:37

Spatial Separation of Molecular Conformers and Clusters

Published on: January 9, 2014

Area of Science:

  • Mathematical Physics
  • Differential Geometry
  • Dynamical Systems

Background:

  • Multiseparable and superintegrable systems are key areas in mathematical physics.
  • Bi-Hamiltonian geometry offers a powerful framework for studying integrable systems.
  • Generalized Lenard chains are a recent development in the theory of integrable systems.

Purpose of the Study:

  • To establish a connection between generalized Lenard chains and the theory of multiseparable and superintegrable systems.
  • To demonstrate that generalized Lenard chains guarantee the separation of variables on specific manifolds.
  • To apply this framework to known physical systems and discover new structures.

Main Methods:

  • Formulation of the theory of multiseparable and superintegrable systems using generalized Lenard chains.
  • Proof of the separation of variables based on the existence of generalized Lenard chains on a four-dimensional manifold.
  • Construction of generalized Lenard chains for the Hénon-Heiles and Smorodinsky-Winternitz systems.
  • Identification of novel bi-Hamiltonian structures for the Kepler potential.

Main Results:

  • Generalized Lenard chains naturally unify the study of multiseparable and superintegrable systems within bi-Hamiltonian geometry.
  • The existence of generalized Lenard chains on a four-dimensional manifold is proven to be a sufficient condition for the separation of variables.
  • Explicit constructions of these chains are provided for the Hénon-Heiles and Smorodinsky-Winternitz systems.
  • New bi-Hamiltonian structures associated with the Kepler potential have been discovered.

Conclusions:

  • Generalized Lenard chains offer a unifying perspective on multiseparable and superintegrable systems.
  • The link between generalized Lenard chains and separation of variables is rigorously established.
  • The study provides new insights and tools for analyzing classical and quantum integrable systems.