哈密爾頓力學 Hamiltonian mechanics
2021-02-24 每天一點純數學哈密爾頓力學是經典力學理論的一個等價但更抽象的重新表述。從歷史上看，它促進了統計力學和量子力學的形成。
Hamiltonian mechanics is an equivalent but more abstract reformulation of classical mechanic theory. Historically, it contributed to the formulation of statistical mechanics and quantum mechanics.
漢密爾頓力學最早由威廉·羅文·漢密爾頓在1833年提出，從拉格朗日力學開始，這是約瑟夫·路易斯·拉格朗日在1788年提出的對經典力學的重新表述。
Hamiltonian mechanics was first formulated by William Rowan Hamilton in 1833, starting from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788.
在哈密頓力學中，一個經典的物理系統是由一組正則坐標r = (q, p)來描述的，其中坐標qi 的每個分量pi都被索引到系統的參照系中。qi 稱為廣義坐標，它的選擇是為了消除約束或利用問題的對稱性，pi是它們的共軛動量。
In Hamiltonian mechanics, a classical physical system is described by a set of canonical coordinates r = (q, p), where each component of the coordinate qi, pi is indexed to the frame of reference of the system.
The qi are called generalized coordinates, and are chosen so as to eliminate the constraints or to take advantage of the symmetries of the problem, and pi are their conjugate momenta.
系統的時間演化由哈密爾頓方程唯一定義:
The time evolution of the system is uniquely defined by Hamilton's equations:
其中H = H(q, p, t)是哈密爾頓量，它通常對應於系統的總能量。對於一個封閉系統，它是系統中動能和勢能的總和。
where H = H(q, p, t) is the Hamiltonian, which often corresponds to the total energy of the system. For a closed system, it is the sum of the kinetic and potential energy in the system.
在牛頓力學中，通過計算系統中每個質點所受的總力得到時間演化，根據牛頓第二定律計算位置和速度的時間演化。而在哈密爾頓力學中，時間演化是通過計算系統在廣義坐標下的哈密爾頓量，並將其插入哈密爾頓方程得到的。這種方法與拉格朗日力學中使用的方法是等價的。哈密爾頓量是拉格朗日函數在q、t固定和p為對偶變量的情況下的勒讓德變換，因此兩種方法對於相同的廣義動量給出了相同的方程。用哈密爾頓力學代替拉格朗日力學的主要動機來自於哈密爾頓系統的辛結構。
In Newtonian mechanics, the time evolution is obtained by computing the total force being exerted on each particle of the system, and from Newton's second law, the time evolutions of both position and velocity are computed.
In contrast, in Hamiltonian mechanics, the time evolution is obtained by computing the Hamiltonian of the system in the generalized coordinates and inserting it into Hamilton's equations. This approach is equivalent to the one used in Lagrangian mechanics.
The Hamiltonian is the Legendre transform of the Lagrangian when holding q and t fixed and defining p as the dual variable, and thus both approaches give the same equations for the same generalized momentum. The main motivation to use Hamiltonian mechanics instead of Lagrangian mechanics comes from the symplectic structure of Hamiltonian systems.
雖然哈密爾頓力學可以用來描述簡單的系統，如彈跳球、鐘擺或振蕩彈簧，在這些系統中，能量從動能轉變為勢能，並隨時間變化，但它的強度表現在更複雜的動態系統中，如天體力學中的行星軌道。系統的自由度越大，它的時間演化就越複雜，在大多數情況下，它就會變得混亂。
While Hamiltonian mechanics can be used to describe simple systems such as a bouncing ball, a pendulum or an oscillating spring in which energy changes from kinetic to potential and back again over time, its strength is shown in more complex dynamic systems, such as planetary orbits in celestial mechanics.
The more degrees of freedom the system has, the more complicated its time evolution is and, in most cases, it becomes chaotic.
基本的物理解釋 Basic physical interpretation哈密爾頓力學的一個簡單解釋來自於它在由一個質量為m的粒子組成的一維系統中的應用。哈密爾頓量可以表示系統的總能量，即動能和勢能的總和，傳統上分別表示為T和V。這裡q是空間坐標，p是動量mv。然後
A simple interpretation of Hamiltonian mechanics comes from its application on a one-dimensional system consisting of one particle of mass m. The Hamiltonian can represent the total energy of the system, which is the sum of kinetic and potential energy, traditionally denoted T and V, respectively. Here q is the space coordinate and p is the momentum mv. Then
T是p單獨的函數，而V是q單獨的函數。， T和V是硬化的)。
T is a function of p alone, while V is a function of q alone (i.e., T and V are scleronomic).
在這個例子中，動量p的時間導數等於牛頓力，所以第一個漢密爾頓方程意味著這個力等於勢能的負梯度。q的時間導數就是速度，所以第二個漢密爾頓方程意味著粒子的速度等於它的動能對動量的導數。
In this example, the time derivative of the momentum p equals the Newtonian force, and so the first Hamilton equation means that the force equals the negative gradient of potential energy.
The time derivative of q is the velocity, and so the second Hamilton equation means that the particle's velocity equals the derivative of its kinetic energy with respect to its momentum.
哈密爾頓系統幾何
哈密爾頓系統可以理解為纖維束E /時間R，其中纖維Et, t ∈ R，是位置空間。拉格朗日函數是射流束J / E上的函數;利用拉格朗日函數的纖維勒讓德變換在雙束上產生一個函數它在t處的纖維是餘切空間T∗Et，它具有一個自然的辛形式，後一個函數是哈密頓函數。拉格朗日力學和哈密頓力學之間的對應關係是用重言式的一種形式實現的。
A Hamiltonian system may be understood as a fiber bundle E over time R, with the fibers Et, t ∈ R, being the position space.
The Lagrangian is thus a function on the jet bundle J over E; taking the fiberwise Legendre transform of the Lagrangian produces a function on the dual bundle over time whose fiber at t is the cotangent space T∗Et, which comes equipped with a natural symplectic form, and this latter function is the Hamiltonian. The correspondence between Lagrangian and Hamiltonian mechanics is achieved with the tautological one-form.
通過泊松括號推廣到量子力學
上面的哈密頓方程對經典力學很適用，但對量子力學就不適用了，因為討論的微分方程假設可以同時指定粒子在任何時間點的確切位置和動量。然而，通過將泊松代數在p和q上的變形推廣到莫雅括號的代數，這些方程可以進一步推廣到量子力學以及經典力學。
Hamilton's equations above work well for classical mechanics, but not for quantum mechanics, since the differential equations discussed assume that one can specify the exact position and momentum of the particle simultaneously at any point in time.
However, the equations can be further generalized to then be extended to apply to quantum mechanics as well as to classical mechanics, through the deformation of the Poisson algebra over p and q to the algebra of Moyal brackets.
具體來說，漢密爾頓方程的更一般形式是這樣的
Specifically, the more general form of the Hamilton's equation reads
其中f是p和q的函數，H是漢密爾頓函數。若要找出不用微分方程求泊松括號的值的規則，請參見李代數;泊松括號是泊松代數中的李括號的名稱。正如Hilbrand J. Groenewold所證明的那樣，這些泊松括號可以擴展到莫雅勒括號，從而描述相空間中的量子力學擴散(參見相空間公式和Wigner-Weyl變換)。這種更代數的方法不僅允許最終擴展相空間中的概率分布到威格納準概率分布，而且，在純粹泊松括號經典設置下，還提供了更大的能力來幫助分析系統中相關的守恆量。
where f is some function of p and q, and H is the Hamiltonian. To find out the rules for evaluating a Poisson bracket without resorting to differential equations, see Lie algebra; a Poisson bracket is the name for the Lie bracket in a Poisson algebra.
These Poisson brackets can then be extended to Moyal brackets comporting to an inequivalent Lie algebra, as proven by Hilbrand J. Groenewold, and thereby describe quantum mechanical diffusion in phase space (See the phase space formulation and the Wigner-Weyl transform).
This more algebraic approach not only permits ultimately extending probability distributions in phase space to Wigner quasi-probability distributions, but, at the mere Poisson bracket classical setting, also provides more power in helping analyze the relevant conserved quantities in a system.
數學形式體系
辛流形上的任何光滑實值函數H都可以用來定義哈密頓系統。函數H被稱為「哈密頓函數」或「能量函數」。這個辛流形被稱為相空間。哈密頓量在辛流形上導出一個特殊的向量場，稱為哈密頓向量場。
Any smooth real-valued function H on a symplectic manifold can be used to define a Hamiltonian system. The function H is known as "the Hamiltonian" or "the energy function." The symplectic manifold is then called the phase space. The Hamiltonian induces a special vector field on the symplectic manifold, known as the Hamiltonian vector field.
哈密爾頓向量場在流形上引起哈密頓流動。這是流形變換的單參數族(曲線的參數通常稱為「時間」);換句話說，是一種共形的同位素，從同一性開始。根據劉維爾定理，每個辛型都保留了相空間上的體積形式。由哈密爾頓流體引起的共形集合通常被稱為哈密爾頓體系的「哈密爾頓力學」。
The Hamiltonian vector field induces a Hamiltonian flow on the manifold. This is a one-parameter family of transformations of the manifold (the parameter of the curves is commonly called "the time"); in other words, an isotopy of symplectomorphisms, starting with the identity.
By Liouville's theorem, each symplectomorphism preserves the volume form on the phase space. The collection of symplectomorphisms induced by the Hamiltonian flow is commonly called "the Hamiltonian mechanics" of the Hamiltonian system.
辛結構產生一個泊松支架。泊松括號給出了李代數流形上的函數空間。
The symplectic structure induces a Poisson bracket. The Poisson bracket gives the space of functions on the manifold the structure of a Lie algebra.
給定一個函數f
Given a function f
如果有一個概率分布,ρ,然後(因為相空間速度(ṗi, q̇i)散度為零和概率是守恆的)其對流導數為零,所以
if there is a probability distribution, ρ, then (since the phase space velocity (ṗi, q̇i) has zero divergence and probability is conserved) its convective derivative can be shown to be zero and so
這就是劉維爾定理。在辛形流形上的每一個光滑函數G生成一個單參數的辛形族，如果{G, H} = 0，則G是守恆的，辛形是對稱變換。
This is called Liouville's theorem. Every smooth function G over the symplectic manifold generates a one-parameter family of symplectomorphisms and if {G, H} = 0, then G is conserved and the symplectomorphisms are symmetry transformations.
哈密爾頓函數可能有多個守恆量Gi。如果辛型流形的維數為2n，且有n個功能獨立的守恆量Gi處於對合狀態(即， {Gi, Gj} = 0)，則哈密爾頓量為劉維爾可積。
A Hamiltonian may have multiple conserved quantities Gi. If the symplectic manifold has dimension 2n and there are n functionally independent conserved quantities Gi which are in involution (i.e., {Gi, Gj} = 0), then the Hamiltonian is Liouville integrable.
劉維爾-阿諾德定理指出，在局部上，任何劉維爾可積哈密爾頓量都可以通過辛型變換轉化為新的哈密爾頓量，其中守恆量Gi為坐標;新的坐標稱為動作角坐標。哈密爾頓量的變換隻依賴於Gi，因此運動方程具有簡單的形式
The Liouville-Arnold theorem says that, locally, any Liouville integrable Hamiltonian can be transformed via a symplectomorphism into a new Hamiltonian with the conserved quantities Gi as coordinates; the new coordinates are called action-angle coordinates.
The transformed Hamiltonian depends only on the Gi, and hence the equations of motion have the simple form
對於函數F (Arnol'd et al.， 1988)。有一個完整的領域專注於可積系統的小偏差，由KAM定理控制。
for some function F (Arnol'd et al., 1988). There is an entire field focusing on small deviations from integrable systems governed by the KAM theorem.
哈密爾頓向量場的可積性是一個懸而未決的問題。一般來說，哈密爾頓系統是混沌的;度量、完全性、可積性和穩定性的概念定義不清。
The integrability of Hamiltonian vector fields is an open question. In general, Hamiltonian systems are chaotic; concepts of measure, completeness, integrability and stability are poorly defined.
黎曼流形
一個重要的特例是由二次型的哈密爾頓式構成的，也就是說，哈密爾頓式可以寫成
An important special case consists of those Hamiltonians that are quadratic forms, that is, Hamiltonians that can be written as
⟨ , ⟩q是一個順利不同纖維T∗
qQ,內積的餘切空間點在配置空間,有時稱為cometric。哈密頓函數完全由動能項組成。
where ⟨ , ⟩q is a smoothly varying inner product on the fibers T∗
qQ, the cotangent space to the point q in the configuration space, sometimes called a cometric. This Hamiltonian consists entirely of the kinetic term.
如果我們考慮一個黎曼流形或一個偽黎曼流形，黎曼度量在正切和餘切叢之間導出一個線性同構。(見音樂同構)。利用這個同構，我們可以定義喜劇。(在坐標中，定義喜劇的矩陣是定義度規的矩陣的倒數。) 哈密爾頓-雅可比方程的解與流形上的測地線相同。特別地，這種情況下的哈密爾頓流和測地線流是一樣的。關於測地線的文章詳細討論了這些解的存在性，以及這組解的完備性。也可以把測地線看作哈密爾頓流。
If one considers a Riemannian manifold or a pseudo-Riemannian manifold, the Riemannian metricinduces a linear isomorphism between the tangent and cotangent bundles. (See Musical isomorphism).
Using this isomorphism, one can define a cometric. (In coordinates, the matrix defining the cometric is the inverse of the matrix defining the metric.)
The solutions to the Hamilton–Jacobi equations for this Hamiltonian are then the same as the geodesics on the manifold. In particular, the Hamiltonian flow in this case is the same thing as the geodesic flow.
The existence of such solutions, and the completeness of the set of solutions, are discussed in detail in the article on geodesics. See also Geodesics as Hamiltonian flows.
亞黎曼流形
當cometric是退化的，那麼它是不可逆的。在這種情況下，我們沒有黎曼流形，因為我們沒有度規。然而，哈密爾頓函數仍然存在。如果在構型空間流形Q的每一點q上，cometric退化，使得cometric的秩小於流形Q的維數，則有一個亞黎曼流形。
When the cometric is degenerate, then it is not invertible. In this case, one does not have a Riemannian manifold, as one does not have a metric. However, the Hamiltonian still exists.
In the case where the cometric is degenerate at every point q of the configuration space manifold Q, so that the rank of the cometric is less than the dimension of the manifold Q, one has a sub-Riemannian manifold.
這種情況下的哈密頓量被稱為亞黎曼哈密爾頓量。每一個這樣的哈密爾頓量都唯一地決定了cometric，反之亦然。這意味著每一個亞黎曼流形都由它的亞黎曼哈密爾頓量唯一決定，反過來也是成立的:每一個亞黎曼流形都有一個唯一的亞黎曼哈密爾頓量。亞黎曼測地線的存在性由喬-拉什夫斯基定理得到。
The Hamiltonian in this case is known as a sub-Riemannian Hamiltonian. Every such Hamiltonian uniquely determines the cometric, and vice versa.
This implies that every sub-Riemannian manifoldis uniquely determined by its sub-Riemannian Hamiltonian, and that the converse is true: every sub-Riemannian manifold has a unique sub-Riemannian Hamiltonian. The existence of sub-Riemannian geodesics is given by the Chow–Rashevskii theorem.
連續實值海森堡群提供了一個關於子亞黎曼流形的簡單例子。對於海森堡群，哈密爾頓量由
The continuous, real-valued Heisenberg group provides a simple example of a sub-Riemannian manifold. For the Heisenberg group, the Hamiltonian is given by
pz與哈密爾頓函數無關。
pz is not involved in the Hamiltonian.
泊松代數
哈密爾頓體系可以用多種方法推廣。哈密爾頓力學不是簡單地研究辛流形上光滑函數的代數，而是可以建立在一般交換么實泊松代數上。狀態是泊松代數上的一個連續線性泛函(具有一些合適的拓撲結構)，這樣對於代數中的任何元素A, A2都映射到一個非負實數。
Hamiltonian systems can be generalized in various ways. Instead of simply looking at the algebra of smooth functions over a symplectic manifold, Hamiltonian mechanics can be formulated on general commutative unital real Poisson algebras.
A state is a continuous linear functional on the Poisson algebra (equipped with some suitable topology) such that for any element A of the algebra, A2maps to a nonnegative real number.
Nambu dynamics給出了進一步的概括。
A further generalization is given by Nambu dynamics.
電磁場中的帶電粒子
通過電磁場中帶電粒子的哈密頓量，給出了哈密頓力學的一個充分說明。在笛卡爾坐標系(即qi = xi)中，非相對論經典粒子在電磁場中的拉格朗日為(SI單位):
A sufficient illustration of Hamiltonian mechanics is given by the Hamiltonian of a charged particle in an electromagnetic field. In Cartesian coordinates (i.e. qi = xi) the Lagrangian of a non-relativistic classical particle in an electromagnetic field is (in SI Units):
e是粒子的電荷(不一定是元電荷),φ是電動標量勢,Ai的組件可能所有明確的磁矢勢取決於xi和t。注意，拉格朗日量本身是量規不變的，但是標量勢和向量勢的值在量規變換過程中會發生變化。
where e is the electric charge of the particle (not necessarily the elementary charge), φ is the electric scalar potential, and the Ai are the components of the magnetic vector potential that may all explicitly depend on xi and t .
Note that the Lagrangian itself is gauge invariant, but the values of scalar potential and vector potential would change during a gauge transformation.
這個拉格朗日方程，結合歐拉-拉格朗日方程，產生了洛倫茲力定律
This Lagrangian, combined with Euler–Lagrange equation, produces the Lorentz force law
稱為最小耦合。
and is called minimal coupling.
廣義動量由以下給出:
The generalized momenta are given by:
重新排列，速度用動量表示
Rearranging, the velocities are expressed in terms of the momenta:
如果把動量的定義，以及動量的速度定義，代入上面給出的哈密頓量的定義，然後進行簡化和重新排列，它被表示為:
If the definition of the momenta is substituted, as well as the definitions of the velocities in terms of the momenta, into the definition of the Hamiltonian given above, and this is then simplified and rearranged, it is rendered as:
這個方程在量子力學中經常使用。
This equation is used frequently in quantum mechanics.
電磁場中的相對論帶電粒子
相對論性帶電粒子的拉格朗日量由:
The Lagrangian for a relativistic charged particle is given by:
因此，粒子的標準(總)動量是
Thus the particle's canonical (total) momentum is
也就是動能和勢能之和。
that is, the sum of the kinetic momentum and the potential momentum.
解出速度，我們得到
Solving for the velocity, we get
所以漢密爾頓函數是
So the Hamiltonian is
這就得到了力方程(相當於歐拉-拉格朗日方程)
This results in the force equation (equivalent to the Euler–Lagrange equation)
我們可以從中推導
from which one can derive
一個等效的表達式哈密頓函數的相對論動量(動能),p = γmẋ(t)，為
An equivalent expression for the Hamiltonian as function of the relativistic (kinetic) momentum, p = γmẋ(t), is
這樣做的好處是p可以用實驗來測量，而P不能。注意,哈密頓(總能量)可以被視為之和相對論能量(動能+其他),E = γmc2,加上勢能,V = eφ。
This has the advantage that p can be measured experimentally whereas P cannot. Notice that the Hamiltonian (total energy) can be viewed as the sum of the relativistic energy (kinetic+rest), E = γmc2, plus the potential energy, V = eφ.
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