English Dictionary

Legendre transformation

名詞

(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.)

日本語の意味

数学におけるルジャンドル変換とは、ある関数 f(x,y,z,...) が変数 x に関して上に凸である（すなわち、x に関する二階微分が正である）場合に、x をその偏微分 p = ∂f/∂x により新たな変数 p と置き換え、F(p,y,z,...) = p·x(p) - f(x(p),y,z,...) という形で定義される新しい関数に変換する手法です。この変換により、元の関数 f の全ての情報が F に含まれることになります。

For a function f that is concave up in x (so ∂2f/∂x2 > 0), performing a Legendre transformation replaces x with its conjugate variable p = ∂f/∂x, yielding a new function that encodes the same information in terms of p.

関数 f が x について凸（すなわち ∂^2f/∂x^2 > 0）である場合、ルジャンドル変換を行うことで x を共役変数 p = ∂f/∂x に置き換え、同じ情報を p の関数として表す新しい関数が得られる。

意味（1）

(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.)

意味（2）

(physics, analytical dynamics) A formula for converting a Lagrangian function to a Hamiltonian function (or vice versa).

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.

意味（3）

(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).

Legendre transformations （ plural ）