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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.)
Definition source: English Wiktionary via Wiktextract
Topics: mathematics, sciences
No example sentence was provided for this meaning.
Meaning relationships
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A formula for converting a Lagrangian function to a Hamiltonian function (or vice versa).
Definition source: English Wiktionary via Wiktextract
Qualifier: analytical dynamics
Topics: natural-sciences, physical-sciences, physics
Examples
- 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.
Meaning relationships
Synonyms: none provided
Antonyms: none provided
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).
Definition source: English Wiktionary via Wiktextract
Topics: natural-sciences, physical-sciences, physics, thermodynamics
No example sentence was provided for this meaning.
Meaning relationships
Synonyms: none provided
Antonyms: none provided
Named after Adrien-Marie Legendre (1752–1852), a French mathematician.
