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Canonical Transformation
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as ''form invariance''. Although Hamilton's equations are preserved, it need not preserve the explicit form of the Hamiltonian itself. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi equations (a useful method for calculating conserved quantities) and Liouville's theorem (itself the basis for classical statistical mechanics). Since Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if the momentum is simultaneously changed by a Legendre transformation into P_i = \frac\ , where \left\ are the new co‑ordinates, grouped in canonical conjugate pairs of momenta P_i and corresponding positions Q_i, for i = 1, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Mechanics
In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta''. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a Hamilton–Jacobi equation, link between classical and quantum mechanics. Overview Phase space coordinates (''p'', ''q'') and Hamiltonian ''H'' Let (M, \mathcal L) be a Lagrangian mechanics, mechanical system with configuration space (physics), configuration space M and smooth Lagrangian_mechanics#Lagrangian, Lagrangian \mathcal L. Select a standard coordinate system (\boldsymbol,\boldsymbol) on M. The quantities \textstyle p_i(\boldsymbol,\boldsymbol,t) ~\stackrel~ / are called ''m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Momentum
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation. Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. This does not mean that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Function (physics)
In physics, and more specifically in Hamiltonian mechanics, a generating function is, loosely, a function whose partial derivatives generate the differential equations that determine a system's dynamics. Common examples are the partition function of statistical mechanics, the Hamiltonian, and the function which acts as a bridge between two sets of canonical variables when performing a canonical transformation. In canonical transformations There are four basic generating functions, summarized by the following table: Example Sometimes a given Hamiltonian can be turned into one that looks like the harmonic oscillator Hamiltonian, which is H = aP^2 + bQ^2. For example, with the Hamiltonian H = \frac + \frac, where is the generalized momentum and is the generalized coordinate, a good canonical transformation to choose would be This turns the Hamiltonian into H = \frac + \frac, which is in the form of the harmonic oscillator Hamiltonian. The generating function for this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Mechanics/Lagrange Theory
Classical may refer to: European antiquity *Classical antiquity, a period of history from roughly the 7th or 8th century B.C.E. to the 5th century C.E. centered on the Mediterranean Sea *Classical architecture, architecture derived from Greek and Roman architecture of classical antiquity *Classical mythology, the body of myths from the ancient Greeks and Romans *Classical tradition, the reception of classical Greco-Roman antiquity by later cultures *Classics, study of the language and culture of classical antiquity, particularly its literature *Classicism, a high regard for classical antiquity in the arts Music and arts *Classical ballet, the most formal of the ballet styles *Classical music, a variety of Western musical styles from the 9th century to the present *Classical guitar, a common type of acoustic guitar *Classical Hollywood cinema, a visual and sound style in the American film industry between 1927 and 1963 *Classical Indian dance, various codified art forms whose theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus Of Variations
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematics), mappings from a set of Function (mathematics), functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''geodesics''. A related problem is posed by Fermat's principle: li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamilton Equations
In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta''. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics. Overview Phase space coordinates (''p'', ''q'') and Hamiltonian ''H'' Let (M, \mathcal L) be a mechanical system with configuration space M and smooth Lagrangian \mathcal L. Select a standard coordinate system (\boldsymbol,\boldsymbol) on M. The quantities \textstyle p_i(\boldsymbol,\boldsymbol,t) ~\stackrel~ / are called ''momenta''. (Also ''generalized momenta'', ''conjugate momenta'', and ''canonical momenta''). For a time instant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Action Integral
Action may refer to: * Action (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 film), a film by Tinto Brass * '' Action 3D'', a 2013 Telugu language film * ''Action'' (2019 film), a Kollywood film. Music * Action (music), a characteristic of a stringed instrument * Action (piano), the mechanism which drops the hammer on the string when a key is pressed * The Action, a 1960s band Albums * ''Action'' (B'z album) (2007) * ''Action!'' (Desmond Dekker album) (1968) * '' Action Action Action'' or ''Action'', a 1965 album by Jackie McLean * ''Action!'' (Oh My God album) (2002) * ''Action'' (Oscar Peterson album) (1968) * ''Action'' (Punchline album) (2004) * ''Action'' (Question Mark & the Mysterians ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Action (physics)
In physics, action is a scalar quantity that describes how the balance of kinetic versus potential energy of a physical system changes with trajectory. Action is significant because it is an input to the principle of stationary action, an approach to classical mechanics that is simpler for multiple objects. Action and the variational principle are used in Feynman's formulation of quantum mechanics and in general relativity. For systems with small values of action close to the Planck constant, quantum effects are significant. In the simple case of a single particle moving with a constant velocity (thereby undergoing uniform linear motion), the action is the momentum of the particle times the distance it moves, added up along its path; equivalently, action is the difference between the particle's kinetic energy and its potential energy, times the duration for which it has that amount of energy. More formally, action is a mathematical functional which takes the trajectory ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Matrix And Determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of components of function values, then its determinant is called the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian. They are named after Carl Gustav Jacob Jacobi. The Jacobian matrix is the natural generalization to vector valued functions of several variables of the derivative and the differential of a usual function. This generalization includes generalizations of the inverse function theorem and the implicit function theorem, where the non-nullity of the derivative is replaced by the non-nullity of the Jacobian determinant, and the multiplicative inverse of the derivative is replaced by the inverse of the Jacobian matrix. The Jacobian determinant is fundamentally use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integration By Substitution
In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards." This involves differential forms. Substitution for a single variable Introduction (indefinite integrals) Before stating the result rigorously, consider a simple case using indefinite integrals. Compute \int(2x^3+1)^7(x^2)\,dx. Set u=2x^3+1. This means \frac=6x^2, or as a differential form, du=6x^2\,dx. Now: \begin \int(2x^3 +1)^7(x^2)\,dx &= \frac\int\underbrace_\underbrace_ \\ &= \frac\int u^\,du \\ &= \frac\left(\fracu^\right)+C \\ &= \frac(2x^3+1)^+C, \end where C is an arbitrary constant of integration. This procedure is frequently used, but not all integrals are of a form that permits its use. In any event, the result should be verified by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Bracket
Lagrange brackets are certain expressions closely related to Poisson brackets that were introduced by Joseph Louis Lagrange in 1808–1810 for the purposes of mathematical formulation of classical mechanics, but unlike the Poisson brackets, have fallen out of use. Definition Suppose that (''q''1, ..., ''q''''n'', ''p''1, ..., ''p''''n'') is a system of canonical coordinates on a phase space. If each of them is expressed as a function of two variables, ''u'' and ''v'', then the Lagrange bracket of ''u'' and ''v'' is defined by the formula : u, v = \sum_^n \left(\frac \frac - \frac \frac \right). Properties * Lagrange brackets do not depend on the system of canonical coordinates (''q'', ''p''). If (''Q'',''P'') = (''Q''1, ..., ''Q''''n'', ''P''1, ..., ''P''''n'') is another system of canonical coordinates, so that Q=Q(q,p), P=P(q,p) is a canonical transformation, then the Lagrange bracket is an invariant of the transformation, in the sense that u, v = , v Ther ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
