"Legendre transformation" meaning in English

{
  "etymology_text": "Named after Adrien-Marie Legendre (1752–1852), a French mathematician.",
  "forms": [
    {
      "form": "Legendre transformations",
      "tags": [
        "plural"
      ]
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  "lang_code": "en",
  "pos": "noun",
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      "glosses": [
        "Given a function f(x,y,z,...) which is concave up with respect to x (i.e., its second derivative with respect to x is greater than zero), an involutive procedure for replacing x with another variable, say p=∂f/∂x thus yielding another function, say F=F(p,y,z,...). This new function contains all of the information of the original f encoded, as it were, within it so that ∂F/∂p=x and applying a similar transformation to F yields the original f. The formula is: F(p,y,z,...)=p·x(p)-f(x(p),y,z,...) where x must be expressed as a function of p. (Note: The concave upwardness means that ∂f/∂x is monotonically increasing, which means that p as a function of x is invertible, so x should be expressible as a function of p.)"
      ],
      "id": "en-Legendre_transformation-en-noun-3gXxX6Rq",
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "involutive",
          "involutive"
        ],
        [
          "monotonically",
          "monotonically"
        ]
      ],
      "raw_glosses": [
        "(mathematics) Given a function f(x,y,z,...) which is concave up with respect to x (i.e., its second derivative with respect to x is greater than zero), an involutive procedure for replacing x with another variable, say p=∂f/∂x thus yielding another function, say F=F(p,y,z,...). This new function contains all of the information of the original f encoded, as it were, within it so that ∂F/∂p=x and applying a similar transformation to F yields the original f. The formula is: F(p,y,z,...)=p·x(p)-f(x(p),y,z,...) where x must be expressed as a function of p. (Note: The concave upwardness means that ∂f/∂x is monotonically increasing, which means that p as a function of x is invertible, so x should be expressible as a function of p.)"
      ],
      "topics": [
        "mathematics",
        "sciences"
      ]
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          "name": "Physics",
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            "All topics",
            "Fundamental"
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        }
      ],
      "examples": [
        {
          "text": "A Legendre transformation looks like this: H=∑ᵢ₌₁ⁿp_i̇q_i-L=∑ᵢ₌₁ⁿ∂L/∂̇q_i̇q_i-L, where the q_i are generalized coordinates, their dotted versions ̇q_i are their time derivatives, the p_i=∂L/∂̇q_i are generalized momenta or conjugate momenta, L=L(q_1,...,q_n;̇q_1,...,̇q_n) is a Lagrangian function and H=H(q_1,...,q_n;p_1,...,p_n) is a Hamiltonian function."
        }
      ],
      "glosses": [
        "A formula for converting a Lagrangian function to a Hamiltonian function (or vice versa)."
      ],
      "id": "en-Legendre_transformation-en-noun-Dfn-dRE7",
      "links": [
        [
          "physics",
          "physics"
        ],
        [
          "Lagrangian function",
          "Lagrangian function"
        ],
        [
          "Hamiltonian function",
          "Hamiltonian function"
        ],
        [
          "vice versa",
          "vice versa"
        ]
      ],
      "qualifier": "analytical dynamics",
      "raw_glosses": [
        "(physics, analytical dynamics) A formula for converting a Lagrangian function to a Hamiltonian function (or vice versa)."
      ],
      "topics": [
        "natural-sciences",
        "physical-sciences",
        "physics"
      ]
    },
    {
      "categories": [
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Thermodynamics",
          "orig": "en:Thermodynamics",
          "parents": [
            "Physics",
            "Sciences",
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            "Fundamental"
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          "source": "w"
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      ],
      "glosses": [
        "A relation between internal energy (expressed in terms of volume and entropy) and enthalpy (replacing volume with pressure), or between internal energy and Helmholtz free energy (replacing entropy with temperature), or between enthalpy and Gibbs free energy (replacing entropy with temperature), or between internal energy and Gibbs free energy (replacing volume with pressure and entropy with temperature), or between Helmholtz free energy and Gibbs free energy (replacing volume with pressure)."
      ],
      "id": "en-Legendre_transformation-en-noun-GUN7c7lH",
      "links": [
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          "thermodynamics",
          "thermodynamics"
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          "relation",
          "relation"
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        [
          "internal energy",
          "internal energy"
        ],
        [
          "entropy",
          "entropy"
        ],
        [
          "enthalpy",
          "enthalpy"
        ],
        [
          "Helmholtz free energy",
          "Helmholtz free energy"
        ],
        [
          "Gibbs free energy",
          "Gibbs free energy"
        ]
      ],
      "raw_glosses": [
        "(thermodynamics) A relation between internal energy (expressed in terms of volume and entropy) and enthalpy (replacing volume with pressure), or between internal energy and Helmholtz free energy (replacing entropy with temperature), or between enthalpy and Gibbs free energy (replacing entropy with temperature), or between internal energy and Gibbs free energy (replacing volume with pressure and entropy with temperature), or between Helmholtz free energy and Gibbs free energy (replacing volume with pressure)."
      ],
      "topics": [
        "natural-sciences",
        "physical-sciences",
        "physics",
        "thermodynamics"
      ]
    }
  ],
  "wikipedia": [
    "Adrien-Marie Legendre",
    "Legendre transformation"
  ],
  "word": "Legendre transformation"
}

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    "Pages with entries"
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  "etymology_text": "Named after Adrien-Marie Legendre (1752–1852), a French mathematician.",
  "forms": [
    {
      "form": "Legendre transformations",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "en:Mathematics"
      ],
      "glosses": [
        "Given a function f(x,y,z,...) which is concave up with respect to x (i.e., its second derivative with respect to x is greater than zero), an involutive procedure for replacing x with another variable, say p=∂f/∂x thus yielding another function, say F=F(p,y,z,...). This new function contains all of the information of the original f encoded, as it were, within it so that ∂F/∂p=x and applying a similar transformation to F yields the original f. The formula is: F(p,y,z,...)=p·x(p)-f(x(p),y,z,...) where x must be expressed as a function of p. (Note: The concave upwardness means that ∂f/∂x is monotonically increasing, which means that p as a function of x is invertible, so x should be expressible as a function of p.)"
      ],
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "involutive",
          "involutive"
        ],
        [
          "monotonically",
          "monotonically"
        ]
      ],
      "raw_glosses": [
        "(mathematics) Given a function f(x,y,z,...) which is concave up with respect to x (i.e., its second derivative with respect to x is greater than zero), an involutive procedure for replacing x with another variable, say p=∂f/∂x thus yielding another function, say F=F(p,y,z,...). This new function contains all of the information of the original f encoded, as it were, within it so that ∂F/∂p=x and applying a similar transformation to F yields the original f. The formula is: F(p,y,z,...)=p·x(p)-f(x(p),y,z,...) where x must be expressed as a function of p. (Note: The concave upwardness means that ∂f/∂x is monotonically increasing, which means that p as a function of x is invertible, so x should be expressible as a function of p.)"
      ],
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        "mathematics",
        "sciences"
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      "categories": [
        "en:Physics"
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      "examples": [
        {
          "text": "A Legendre transformation looks like this: H=∑ᵢ₌₁ⁿp_i̇q_i-L=∑ᵢ₌₁ⁿ∂L/∂̇q_i̇q_i-L, where the q_i are generalized coordinates, their dotted versions ̇q_i are their time derivatives, the p_i=∂L/∂̇q_i are generalized momenta or conjugate momenta, L=L(q_1,...,q_n;̇q_1,...,̇q_n) is a Lagrangian function and H=H(q_1,...,q_n;p_1,...,p_n) is a Hamiltonian function."
        }
      ],
      "glosses": [
        "A formula for converting a Lagrangian function to a Hamiltonian function (or vice versa)."
      ],
      "links": [
        [
          "physics",
          "physics"
        ],
        [
          "Lagrangian function",
          "Lagrangian function"
        ],
        [
          "Hamiltonian function",
          "Hamiltonian function"
        ],
        [
          "vice versa",
          "vice versa"
        ]
      ],
      "qualifier": "analytical dynamics",
      "raw_glosses": [
        "(physics, analytical dynamics) A formula for converting a Lagrangian function to a Hamiltonian function (or vice versa)."
      ],
      "topics": [
        "natural-sciences",
        "physical-sciences",
        "physics"
      ]
    },
    {
      "categories": [
        "en:Thermodynamics"
      ],
      "glosses": [
        "A relation between internal energy (expressed in terms of volume and entropy) and enthalpy (replacing volume with pressure), or between internal energy and Helmholtz free energy (replacing entropy with temperature), or between enthalpy and Gibbs free energy (replacing entropy with temperature), or between internal energy and Gibbs free energy (replacing volume with pressure and entropy with temperature), or between Helmholtz free energy and Gibbs free energy (replacing volume with pressure)."
      ],
      "links": [
        [
          "thermodynamics",
          "thermodynamics"
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        [
          "relation",
          "relation"
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        [
          "internal energy",
          "internal energy"
        ],
        [
          "entropy",
          "entropy"
        ],
        [
          "enthalpy",
          "enthalpy"
        ],
        [
          "Helmholtz free energy",
          "Helmholtz free energy"
        ],
        [
          "Gibbs free energy",
          "Gibbs free energy"
        ]
      ],
      "raw_glosses": [
        "(thermodynamics) A relation between internal energy (expressed in terms of volume and entropy) and enthalpy (replacing volume with pressure), or between internal energy and Helmholtz free energy (replacing entropy with temperature), or between enthalpy and Gibbs free energy (replacing entropy with temperature), or between internal energy and Gibbs free energy (replacing volume with pressure and entropy with temperature), or between Helmholtz free energy and Gibbs free energy (replacing volume with pressure)."
      ],
      "topics": [
        "natural-sciences",
        "physical-sciences",
        "physics",
        "thermodynamics"
      ]
    }
  ],
  "wikipedia": [
    "Adrien-Marie Legendre",
    "Legendre transformation"
  ],
  "word": "Legendre transformation"
}