• Thumbnail for Legendre transformation
    In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface...
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  • Thumbnail for Adrien-Marie Legendre
    mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him. He is also known for his contributions...
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  • of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation, or Fenchel...
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  • Hamilton's equations if the momentum is simultaneously changed by a Legendre transformation into P i = ∂ L ∂ Q ˙ i   , {\displaystyle P_{i}={\frac {\partial...
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  • Thumbnail for Legendre polynomials
    In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number...
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  • Associated Legendre polynomials Legendre's equation Legendre polynomials Legendre symbol Legendre transformation Legendre (crater), a lunar impact crater...
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  • Thumbnail for Young's inequality for products
    Legendre transformation Integral of inverse functions – Mathematical theorem, used in calculus Legendre transformation – Mathematical transformation Young's...
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  • Thumbnail for Lagrangian mechanics
    L}{\partial {\dot {q}}_{i}}}-L} and can be obtained by performing a Legendre transformation on the Lagrangian, which introduces new variables canonically conjugate...
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  • Thumbnail for Contact geometry
    aims of studying differential equations (e.g. the Legendre transformation or canonical transformation) and describing the 'change of space element', familiar...
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  • momenta in phase space). Both formulations are equivalent by a Legendre transformation on the generalized coordinates, velocities and momenta; therefore...
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  • Thumbnail for Convex hull
    Convex metric space Convex optimization Duality Lagrange multiplier Legendre transformation Locally convex topological vector space Simplex Maps Convex conjugate...
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  • Thumbnail for Hamiltonian mechanics
    canonical momenta). For a time instant t , {\displaystyle t,} the Legendre transformation of L {\displaystyle {\mathcal {L}}} is defined as the map ( q ...
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  • In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation ( 1 − x 2 ) d 2 d x 2 P ℓ m ( x ) − 2...
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  • relation Legendre sieve Legendre symbol Legendre transformation Legendre transform (integral transform) Finite Legendre transform Legendre wavelet Legendre–Clebsch...
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  • Whipple's transformation for Legendre functions, named after Francis John Welsh Whipple, arise from a general expression, concerning associated Legendre functions...
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  • change of thermodynamic potential is equivalent to a Legendre transformation, and Legendre transformations do not act in the equilibrium space, it is necessary...
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  • Thumbnail for Non-equilibrium thermodynamics
    thermodynamic potential Helmholtz free energy (A = U - TS), a Legendre transformation of the energy. If, next to fluctuations of the energy, the macroscopic...
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  • where C {\displaystyle C} is a constant. The left hand side is the Legendre transformation of L {\displaystyle L} with respect to f ′ ( x ) . {\displaystyle...
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  • Functional Theory – Lecture Notes Density Functional Theory through Legendre Transformation Archived 2010-05-10 at the Wayback Machinepdf Burke, Kieron. "The...
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  • (p1, p2, ..., pn) is an n-tuple of the generalized momenta. A Legendre transformation is performed to change the variables in the total differential...
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  • integral Geodesic Isoperimetry Lagrangian Lagrangian mechanics Legendre transformation Luke's variational principle Minimal surface Morse theory Noether's...
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  • involving their quadratic covariation. Integration by substitution Legendre transformation "Brook Taylor". History.MCS.St-Andrews.ac.uk. Retrieved May 25...
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  • (digital image processing) Inverse scattering transform Legendre transformation Möbius transformation Perspective transform (computer graphics) Sequence transform...
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  • The action S {\displaystyle S} in Hamilton's principle is the Legendre transformation of the action in Maupertuis' principle. The concepts and many of...
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  • Thumbnail for Quantum field theory
    spatial and time coordinates, obtained from the Hamiltonian H via Legendre transformation. The initial and final conditions of the path integral are respectively...
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  • Thumbnail for Involution (mathematics)
    resulting in the form (B, G, R): f(f(RGB)) = RGB, f(f(BGR)) = BGR. Legendre transformation, which converts between the Lagrangian and Hamiltonian, is an involutive...
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  • Thumbnail for Classical mechanics
    the configuration space). Both formulations are equivalent by a Legendre transformation on the generalized coordinates, velocities and momenta; therefore...
    52 KB (5,857 words) - 21:39, 15 May 2025
  • Lagrangian stage, before passing over to Hamiltonian mechanics via the Legendre transformation. The Hamiltonian density is related to the Lie derivative of the...
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  • Lagrangian formalism by a Legendre transformation, or from another set of canonical coordinates by a canonical transformation. Canonical coordinates are...
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  • Thumbnail for Effective action
    correlation functions. The quantum effective action is defined using the Legendre transformation of W [ J ] {\displaystyle W[J]} Γ [ ϕ ] = W [ J ϕ ] − ∫ d 4 x J...
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