Hamilton–Jacobi equation
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In mathematics, the Hamilton–Jacobi equation is a necessary condition describing extremalgeometry in generalizations of calculus-of-variations problems. In physics, the Hamilton–Jacobi equation (HJE) is a reformulation of classical mechanics and, thus, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics真善美的反义词. The Hamilton–Jacobi equation is particularly uful in identifying conrved quantities for mechanical systems, which may be possible even when the mechanical problem itlf cannot be solved completely.
The HJE is also the only formulation of mechanics in which the motion of a particle can be reprented as a wave. In this n, the HJE fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the 18th century) of finding an analogy between the propagation of light and the motion of a particle. The wave equation followed by mechanical systems is similar to, but not identical with, 可怕的近义词Schrödinger's equation, as descri
bed below; for this reason, the HJE is considered the "clost approach" of classical mechanics to quantum mechanics.蔡伦是哪个朝代的[1][2]
梅菜红烧肉Mathematical formulation
The Hamilton–Jacobi equation is a first-order, non-linear partial differential equation for a functioncalled Hamilton's principal function
As described below, this equation may be derived from Hamiltonian mechanics by treating S as the generating function for a canonical transformation of the classical Hamiltonian
.
The conjugate momenta correspond to the first derivatives of S with respect to the generalized coordinates
Principal function as solved from the equation from contains N+1 undetermined constants, the last being one from integrating , and the first N denoted as . The relationship then between p and q兔子新娘 describes the orbit in pha space in terms of the constants of motion, and
are also constants of motion and can be inverted to solve q违反唯一约束条件.
Comparison with other formulations of mechanics
The HJE is a single, first-order partial differential equation for the function S of the Ngeneralized coordinatesand the time t. The generalized momenta do not appear, except as derivatives of S. Remarkably, the function S is equal to the classical action.
For comparison, in the equivalent Euler–Lagrange equations of motion of Lagrangian mechanics, the conjugate momenta also do not appear; however, tho equations are a system of N, generally cond-order equations for the time evolution of the generalized coordinates. Similarly, Hamilton's equations of motion are another system of 2N first-order equations for the time evolution of the generalized coordinates and their conjugate momenta .
Since the HJE is an equivalent expression of an integral minimization problem such as Hamilton's principle, the HJE can be uful in other problems of the calculus of variations and, more generally, in other branches of mathematics and physics, such as dynamical systems, symplectic geometry and quantum chaos. For example, the Hamilton–Jacobi equations can be ud to determine the geodesics on a Riemannian manifold, an important variational problem in Riemannian geometry.
Notation
For brevity, we u boldface variables such as to reprent the list of Ngeneralized coordinates
that need not transform like a vector under rotation. The dot product is defined here as the sum of the products of corresponding components, i.e.,
Derivation
Any canonical transformation involving a type-2 generating function leads to the relations
(See the canonical transformation article for more details.)
To derive the HJE, we choo a generating function that makes the new Hamiltonian K identically zero. Hence, all its derivatives are also zero, and Hamilton's equations become trivial
i.e., the new generalized coordinates and momenta are constants of motion. The new generalized momenta are usually denoted , i.e., Pm = αm.
The equation for the transformed Hamiltonian K
Let
whereA is a arbitrary constant, then S satisfies HJE
since .
The new generalized coordinatesare also constants, typically denoted as . Once we have solved for, the also give uful equations
or written in components for clarity
Ideally, the N equations can be inverted to find the original generalized coordinatesas a function of the constants and , thus solving the original problem.
Action
Both Hamilton principal function S and characteristic function are cloly related to action.
The time derivative of S is
therefore
so S is actually classical action plus an undetermined constant.When H金约 does not explicitly depend on time,
风雨任平生in this ca W is the same as abbreviated action.
Separation of variables
The HJE is most uful when it can be solved via additive paration of variables, which directly identifies constants of motion. For example, the time t can be parated if the Hamiltonian does not depend on time explicitly. In that ca, the time derivativein the HJE must be a constant (usually denoted − E), giving the parated solution
