Misplaced Pages

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84-404: The Ising model was invented by the physicist Wilhelm Lenz  ( 1920 ), who gave it as a problem to his student Ernst Ising. The one-dimensional Ising model was solved by Ising (1925) alone in his 1924 thesis; it has no phase transition. The two-dimensional square-lattice Ising model is much harder and was only given an analytic description much later, by Lars Onsager  ( 1944 ). It

168-911: A b ( ∑ i = 1 n p i q ˙ i − H ( p , q , t ) ) d t , {\displaystyle {\mathcal {S}}[{\boldsymbol {q}}]=\int _{a}^{b}{\mathcal {L}}(t,{\boldsymbol {q}}(t),{\dot {\boldsymbol {q}}}(t))\,dt=\int _{a}^{b}\left(\sum _{i=1}^{n}p_{i}{\dot {q}}^{i}-{\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}},t)\right)\,dt,} where ⁠ q = q ( t ) {\displaystyle {\boldsymbol {q}}={\boldsymbol {q}}(t)} ⁠ , and p = ∂ L / ∂ q ˙ {\displaystyle {\boldsymbol {p}}=\partial {\mathcal {L}}/\partial {\boldsymbol {\dot {q}}}} (see above). A path q ∈ P (

252-520: A , b , x a , x b ) {\displaystyle {\boldsymbol {q}}\in {\mathcal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})} is a stationary point of S {\displaystyle {\mathcal {S}}} (and hence is an equation of motion) if and only if the path ( p ( t ) , q ( t ) ) {\displaystyle ({\boldsymbol {p}}(t),{\boldsymbol {q}}(t))} in phase space coordinates obeys

336-671: A , b , x a , x b ) {\displaystyle {\mathcal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})} be the set of smooth paths q : [ a , b ] → M {\displaystyle {\boldsymbol {q}}:[a,b]\to M} for which q ( a ) = x a {\displaystyle {\boldsymbol {q}}(a)={\boldsymbol {x}}_{a}} and q ( b ) = x b . {\displaystyle {\boldsymbol {q}}(b)={\boldsymbol {x}}_{b}.} The action functional S : P (

420-450: A , b , x a , x b ) → R {\displaystyle {\mathcal {S}}:{\mathcal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})\to \mathbb {R} } is defined via S [ q ] = ∫ a b L ( t , q ( t ) , q ˙ ( t ) ) d t = ∫

504-1657: A , b , c ) ∂ c = 0 {\displaystyle {\frac {\partial f(a,b,c)}{\partial c}}=0} . Starting from definitions of the Hamiltonian, generalized momenta, and Lagrangian for an n {\displaystyle n} degrees of freedom system H = ∑ i = 1 n ( p i q ˙ i ) − L ( q , q ˙ , t ) {\displaystyle {\mathcal {H}}=\sum _{i=1}^{n}{\biggl (}p_{i}{\dot {q}}_{i}{\biggr )}-{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)} p i ( q , q ˙ , t ) = ∂ L ( q , q ˙ , t ) ∂ q ˙ i {\displaystyle p_{i}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)={\frac {\partial {\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)}{\partial {\dot {q}}_{i}}}} L ( q , q ˙ , t ) = T ( q , q ˙ , t ) − V ( q , q ˙ , t ) {\displaystyle {\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)=T({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)-V({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)} Substituting

588-475: A cyclic coordinate ), the corresponding momentum coordinate p i {\displaystyle p_{i}} is conserved along each trajectory, and that coordinate can be reduced to a constant in the other equations of the set. This effectively reduces the problem from n coordinates to ( n − 1) coordinates: this is the basis of symplectic reduction in geometry. In the Lagrangian framework,

672-799: A mass m moving without friction on the surface of a sphere . The only forces acting on the mass are the reaction from the sphere and gravity . Spherical coordinates are used to describe the position of the mass in terms of ( r , θ , φ ) , where r is fixed, r = ℓ . The Lagrangian for this system is L = 1 2 m ℓ 2 ( θ ˙ 2 + sin 2 ⁡ θ   φ ˙ 2 ) + m g ℓ cos ⁡ θ . {\displaystyle L={\frac {1}{2}}m\ell ^{2}\left({\dot {\theta }}^{2}+\sin ^{2}\theta \ {\dot {\varphi }}^{2}\right)+mg\ell \cos \theta .} Thus

756-1196: A function H ( p , q , t ) {\displaystyle {\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}},t)} known as the Hamiltonian . The Hamiltonian satisfies H ( ∂ L ∂ q ˙ , q , t ) = E L ( q , q ˙ , t ) {\displaystyle {\mathcal {H}}\left({\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {\dot {q}}}}},{\boldsymbol {q}},t\right)=E_{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)} which implies that H ( p , q , t ) = ∑ i = 1 n p i q ˙ i − L ( q , q ˙ , t ) , {\displaystyle {\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}},t)=\sum _{i=1}^{n}p_{i}{\dot {q}}^{i}-{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t),} where

840-483: A new, unusual phase transition behavior, along with non-vanishing long-range and nearest-neighbor spin-spin correlations, deemed relevant to large neural networks as one of its possible applications . The Ising problem without an external field can be equivalently formulated as a graph maximum cut (Max-Cut) problem that can be solved via combinatorial optimization . Consider a set Λ {\displaystyle \Lambda } of lattice sites, each with

924-500: A set of adjacent sites (e.g. a graph ) forming a d {\displaystyle d} -dimensional lattice. For each lattice site k ∈ Λ {\displaystyle k\in \Lambda } there is a discrete variable σ k {\displaystyle \sigma _{k}} such that σ k ∈ { − 1 , + 1 } {\displaystyle \sigma _{k}\in \{-1,+1\}} , representing

SECTION 10

#1732791669306

1008-757: A standard coordinate system ( q , q ˙ ) {\displaystyle ({\boldsymbol {q}},{\boldsymbol {\dot {q}}})} on M . {\displaystyle M.} The quantities p i ( q , q ˙ , t )   = def   ∂ L / ∂ q ˙ i {\displaystyle \textstyle p_{i}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)~{\stackrel {\text{def}}{=}}~{\partial {\mathcal {L}}}/{\partial {\dot {q}}^{i}}} are called momenta . (Also generalized momenta , conjugate momenta , and canonical momenta ). For

1092-443: A system of point masses, the requirement for T {\displaystyle T} to be quadratic in generalised velocity is always satisfied for the case where T ( q , q ˙ , t ) = T ( q , q ˙ ) {\displaystyle T({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)=T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})} , which

1176-493: A time instant t , {\displaystyle t,} the Legendre transformation of L {\displaystyle {\mathcal {L}}} is defined as the map ( q , q ˙ ) → ( p , q ) {\displaystyle ({\boldsymbol {q}},{\boldsymbol {\dot {q}}})\to \left({\boldsymbol {p}},{\boldsymbol {q}}\right)} which

1260-754: A trajectory in phase space with velocities ⁠ q ˙ i = d d t q i ( t ) {\displaystyle {\dot {q}}^{i}={\tfrac {d}{dt}}q^{i}(t)} ⁠ , obeying Lagrange's equations : d d t ∂ L ∂ q ˙ i − ∂ L ∂ q i = 0   . {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}=0\ .} Rearranging and writing in terms of

1344-1396: A weight of the edge i j {\displaystyle ij} and the scaling 1/2 is introduced to compensate for double counting the same weights W i j = W j i {\displaystyle W_{ij}=W_{ji}} . The identities H ( σ ) = − ∑ i j ∈ E ( V + ) J i j − ∑ i j ∈ E ( V − ) J i j + ∑ i j ∈ δ ( V + ) J i j = − ∑ i j ∈ E ( G ) J i j + 2 ∑ i j ∈ δ ( V + ) J i j , {\displaystyle {\begin{aligned}H(\sigma )&=-\sum _{ij\in E(V^{+})}J_{ij}-\sum _{ij\in E(V^{-})}J_{ij}+\sum _{ij\in \delta (V^{+})}J_{ij}\\&=-\sum _{ij\in E(G)}J_{ij}+2\sum _{ij\in \delta (V^{+})}J_{ij},\end{aligned}}} where

1428-680: Is scleronomic ), V {\displaystyle V} does not contain generalised velocity as an explicit variable, and each term of T {\displaystyle T} is quadratic in generalised velocity. Preliminary to this proof, it is important to address an ambiguity in the related mathematical notation. While a change of variables can be used to equate L ( p , q , t ) = L ( q , q ˙ , t ) {\displaystyle {\mathcal {L}}({\boldsymbol {p}},{\boldsymbol {q}},t)={\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)} , it

1512-948: Is a requirement for H = T + V {\displaystyle {\mathcal {H}}=T+V} anyway. Consider the kinetic energy for a system of N point masses. If it is assumed that T ( q , q ˙ , t ) = T ( q , q ˙ ) {\displaystyle T({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)=T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})} , then it can be shown that r ˙ k ( q , q ˙ , t ) = r ˙ k ( q , q ˙ ) {\displaystyle {\dot {\mathbf {r} }}_{k}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)={\dot {\mathbf {r} }}_{k}({\boldsymbol {q}},{\boldsymbol {\dot {q}}})} (See Scleronomous § Application ). Therefore,

1596-1295: Is a result of Euler's homogeneous function theorem . Hence, the Hamiltonian becomes H = ∑ i = 1 n ( ∂ T ( q , q ˙ ) ∂ q ˙ i q ˙ i ) − T ( q , q ˙ ) + V ( q , t ) = 2 T ( q , q ˙ ) − T ( q , q ˙ ) + V ( q , t ) = T ( q , q ˙ ) + V ( q , t ) {\displaystyle {\begin{aligned}{\mathcal {H}}&=\sum _{i=1}^{n}\left({\frac {\partial T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})}{\partial {\dot {q}}_{i}}}{\dot {q}}_{i}\right)-T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})+V({\boldsymbol {q}},t)\\&=2T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})-T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})+V({\boldsymbol {q}},t)\\&=T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})+V({\boldsymbol {q}},t)\end{aligned}}} For

1680-392: Is an interaction J i j {\displaystyle J_{ij}} . Also a site j ∈ Λ {\displaystyle j\in \Lambda } has an external magnetic field h j {\displaystyle h_{j}} interacting with it. The energy of a configuration σ {\displaystyle {\sigma }}

1764-1728: Is an arbitrary scalar function of q {\displaystyle {\boldsymbol {q}}} . Differentiating this with respect to q ˙ l {\displaystyle {\dot {q}}_{l}} , l ∈ [ 1 , n ] {\displaystyle l\in [1,n]} , gives ∂ T ( q , q ˙ ) ∂ q ˙ l = ∑ i = 1 n ∑ j = 1 n ( ∂ [ c i j ( q ) q ˙ i q ˙ j ] ∂ q ˙ l ) = ∑ i = 1 n ∑ j = 1 n ( c i j ( q ) ∂ [ q ˙ i q ˙ j ] ∂ q ˙ l ) {\displaystyle {\begin{aligned}{\frac {\partial T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})}{\partial {\dot {q}}_{l}}}&=\sum _{i=1}^{n}\sum _{j=1}^{n}{\biggl (}{\frac {\partial \left[c_{ij}({\boldsymbol {q}}){\dot {q}}_{i}{\dot {q}}_{j}\right]}{\partial {\dot {q}}_{l}}}{\biggr )}\\&=\sum _{i=1}^{n}\sum _{j=1}^{n}{\biggl (}c_{ij}({\boldsymbol {q}}){\frac {\partial \left[{\dot {q}}_{i}{\dot {q}}_{j}\right]}{\partial {\dot {q}}_{l}}}{\biggr )}\end{aligned}}} Splitting

SECTION 20

#1732791669306

1848-1163: Is assumed to have a smooth inverse ( p , q ) → ( q , q ˙ ) . {\displaystyle ({\boldsymbol {p}},{\boldsymbol {q}})\to ({\boldsymbol {q}},{\boldsymbol {\dot {q}}}).} For a system with n {\displaystyle n} degrees of freedom, the Lagrangian mechanics defines the energy function E L ( q , q ˙ , t ) = def ∑ i = 1 n q ˙ i ∂ L ∂ q ˙ i − L . {\displaystyle E_{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)\,{\stackrel {\text{def}}{=}}\,\sum _{i=1}^{n}{\dot {q}}^{i}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}-{\mathcal {L}}.} The Legendre transform of L {\displaystyle {\mathcal {L}}} turns E L {\displaystyle E_{\mathcal {L}}} into

1932-462: Is further simplified to H ( σ ) = − J ∑ ⟨ i   j ⟩ σ i σ j . {\displaystyle H(\sigma )=-J\sum _{\langle i~j\rangle }\sigma _{i}\sigma _{j}.} A subset S of the vertex set V(G) of a weighted undirected graph G determines a cut of the graph G into S and its complementary subset G\S. The size of

2016-523: Is given by the Hamiltonian function H ( σ ) = − ∑ ⟨ i j ⟩ J i j σ i σ j − μ ∑ j h j σ j , {\displaystyle H(\sigma )=-\sum _{\langle ij\rangle }J_{ij}\sigma _{i}\sigma _{j}-\mu \sum _{j}h_{j}\sigma _{j},} where

2100-601: Is important to note that ∂ L ( q , q ˙ , t ) ∂ q ˙ i ≠ ∂ L ( p , q , t ) ∂ q ˙ i {\displaystyle {\frac {\partial {\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)}{\partial {\dot {q}}_{i}}}\neq {\frac {\partial {\mathcal {L}}({\boldsymbol {p}},{\boldsymbol {q}},t)}{\partial {\dot {q}}_{i}}}} . In this case,

2184-424: Is increasing with respect to any set of coupling constants J B {\displaystyle J_{B}} . The Simon-Lieb inequality states that for any set S {\displaystyle S} disconnecting x {\displaystyle x} from y {\displaystyle y} (e.g. the boundary of a box with x {\displaystyle x} being inside

2268-1174: Is not a derivative of ⁠ q i {\displaystyle q^{i}} ⁠ ). The total differential of the Lagrangian is: d L = ∑ i ( ∂ L ∂ q i d q i + ∂ L ∂ q ˙ i d q ˙ i ) + ∂ L ∂ t d t   . {\displaystyle \mathrm {d} {\mathcal {L}}=\sum _{i}\left({\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}\,\mathrm {d} {\dot {q}}^{i}\right)+{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\ .} The generalized momentum coordinates were defined as ⁠ p i = ∂ L / ∂ q ˙ i {\displaystyle p_{i}=\partial {\mathcal {L}}/\partial {\dot {q}}^{i}} ⁠ , so we may rewrite

2352-622: Is of the form T ( q , q ˙ ) = ∑ i = 1 n ∑ j = 1 n ( c i j ( q ) q ˙ i q ˙ j ) {\displaystyle T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})=\sum _{i=1}^{n}\sum _{j=1}^{n}{\biggl (}c_{ij}({\boldsymbol {q}}){\dot {q}}_{i}{\dot {q}}_{j}{\biggr )}} where each c i j ( q ) {\displaystyle c_{ij}({\boldsymbol {q}})}

2436-411: Is still often assumed that "Ising model" means a ferromagnetic Ising model. In a ferromagnetic Ising model, spins desire to be aligned: the configurations in which adjacent spins are of the same sign have higher probability. In an antiferromagnetic model, adjacent spins tend to have opposite signs. The sign convention of H (σ) also explains how a spin site j interacts with the external field. Namely,

2520-1394: Is the Legendre transform of the Lagrangian L ( q , q ˙ ) {\displaystyle {\mathcal {L}}({\boldsymbol {q}},{\dot {\boldsymbol {q}}})} , thus one has L ( q , q ˙ ) + H ( p , q ) = p q ˙ {\displaystyle {\mathcal {L}}({\boldsymbol {q}},{\dot {\boldsymbol {q}}})+{\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}})={\boldsymbol {p}}{\dot {\boldsymbol {q}}}} and thus ∂ H ∂ p = q ˙ ∂ L ∂ q = − ∂ H ∂ q , {\displaystyle {\begin{aligned}{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {p}}}}&={\dot {\boldsymbol {q}}}\\{\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {q}}}}&=-{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}},\end{aligned}}} Besides, since p = ∂ L / ∂ q ˙ {\displaystyle {\boldsymbol {p}}=\partial {\mathcal {L}}/\partial {\dot {\boldsymbol {q}}}} ,

2604-411: Is the momentum mv and q is the space coordinate. Then H = T + V , T = p 2 2 m , V = V ( q ) {\displaystyle {\mathcal {H}}=T+V,\qquad T={\frac {p^{2}}{2m}},\qquad V=V(q)} T is a function of p alone, while V is a function of q alone (i.e., T and V are scleronomic ). In this example,

Ising model - Misplaced Pages Continue

2688-565: Is time, n {\displaystyle n} is the number of degrees of freedom of the system, and each c i j ( q ) {\displaystyle c_{ij}({\boldsymbol {q}})} is an arbitrary scalar function of q {\displaystyle {\boldsymbol {q}}} . In words, this means that the relation H = T + V {\displaystyle {\mathcal {H}}=T+V} holds true if T {\displaystyle T} does not contain time as an explicit variable (it

2772-545: Is uniquely solvable for ⁠ q ˙ {\displaystyle {\boldsymbol {\dot {q}}}} ⁠ . The ( 2 n {\displaystyle 2n} -dimensional) pair ( p , q ) {\displaystyle ({\boldsymbol {p}},{\boldsymbol {q}})} is called phase space coordinates . (Also canonical coordinates ). In phase space coordinates ⁠ ( p , q ) {\displaystyle ({\boldsymbol {p}},{\boldsymbol {q}})} ⁠ ,

2856-660: Is usually solved by a transfer-matrix method , although there exist different approaches, more related to quantum field theory . In dimensions greater than four, the phase transition of the Ising model is described by mean-field theory . The Ising model for greater dimensions was also explored with respect to various tree topologies in the late 1970s, culminating in an exact solution of the zero-field, time-independent Barth (1981) model for closed Cayley trees of arbitrary branching ratio, and thereby, arbitrarily large dimensionality within tree branches. The solution to this model exhibited

2940-415: The σ {\displaystyle \sigma } -depended set of edges that connects the two complementary vertex subsets V + {\displaystyle V^{+}} and V − {\displaystyle V^{-}} . The size | δ ( V + ) | {\displaystyle \left|\delta (V^{+})\right|} of

3024-498: The path integral formulation and the Schrödinger equation . In its application to a given system, the Hamiltonian is often taken to be H = T + V {\displaystyle {\mathcal {H}}=T+V} where T {\displaystyle T} is the kinetic energy and V {\displaystyle V} is the potential energy. Using this relation can be simpler than first calculating

3108-456: The spontaneous magnetization for the 2-dimensional model in 1949 but did not give a derivation. Yang (1952) gave the first published proof of this formula, using a limit formula for Fredholm determinants , proved in 1951 by Szegő in direct response to Onsager's work. A number of correlation inequalities have been derived rigorously for the Ising spin correlations (for general lattice structures), which have enabled mathematicians to study

3192-1260: The ( n {\displaystyle n} -dimensional) Euler–Lagrange equation ∂ L ∂ q − d d t ∂ L ∂ q ˙ = 0 {\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {q}}}}-{\frac {d}{dt}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {\boldsymbol {q}}}}}=0} becomes Hamilton's equations in 2 n {\displaystyle 2n} dimensions d q d t = ∂ H ∂ p , d p d t = − ∂ H ∂ q . {\displaystyle {\frac {\mathrm {d} {\boldsymbol {q}}}{\mathrm {d} t}}={\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {p}}}},\quad {\frac {\mathrm {d} {\boldsymbol {p}}}{\mathrm {d} t}}=-{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}}.} The Hamiltonian H ( p , q ) {\displaystyle {\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}})}

3276-606: The Euler–Lagrange equations yield p ˙ = d p d t = ∂ L ∂ q = − ∂ H ∂ q . {\displaystyle {\dot {\boldsymbol {p}}}={\frac {\mathrm {d} {\boldsymbol {p}}}{\mathrm {d} t}}={\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {q}}}}=-{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}}.} Let P (

3360-498: The Hamilton's equations. A simple interpretation of Hamiltonian mechanics comes from its application on a one-dimensional system consisting of one nonrelativistic particle of mass m . The value H ( p , q ) {\displaystyle H(p,q)} of the Hamiltonian is the total energy of the system, in this case the sum of kinetic and potential energy , traditionally denoted T and V , respectively. Here p

3444-429: The Hamiltonian function H ( σ ) {\displaystyle H(\sigma )} is conventional. Using this sign convention, Ising models can be classified according to the sign of the interaction: if, for a pair i ,  j The system is called ferromagnetic or antiferromagnetic if all interactions are ferromagnetic or all are antiferromagnetic. The original Ising models were ferromagnetic, and it

Ising model - Misplaced Pages Continue

3528-1022: The Hamiltonian is H = P θ θ ˙ + P φ φ ˙ − L {\displaystyle H=P_{\theta }{\dot {\theta }}+P_{\varphi }{\dot {\varphi }}-L} where P θ = ∂ L ∂ θ ˙ = m ℓ 2 θ ˙ {\displaystyle P_{\theta }={\frac {\partial L}{\partial {\dot {\theta }}}}=m\ell ^{2}{\dot {\theta }}} and P φ = ∂ L ∂ φ ˙ = m ℓ 2 sin 2 θ φ ˙ . {\displaystyle P_{\varphi }={\frac {\partial L}{\partial {\dot {\varphi }}}}=m\ell ^{2}\sin ^{2}\!\theta \,{\dot {\varphi }}.} In terms of coordinates and momenta,

3612-1140: The Hamiltonian reads H = [ 1 2 m ℓ 2 θ ˙ 2 + 1 2 m ℓ 2 sin 2 θ φ ˙ 2 ] ⏟ T + [ − m g ℓ cos ⁡ θ ] ⏟ V = P θ 2 2 m ℓ 2 + P φ 2 2 m ℓ 2 sin 2 ⁡ θ − m g ℓ cos ⁡ θ . {\displaystyle H=\underbrace {\left[{\frac {1}{2}}m\ell ^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}m\ell ^{2}\sin ^{2}\!\theta \,{\dot {\varphi }}^{2}\right]} _{T}+\underbrace {{\Big [}-mg\ell \cos \theta {\Big ]}} _{V}={\frac {P_{\theta }^{2}}{2m\ell ^{2}}}+{\frac {P_{\varphi }^{2}}{2m\ell ^{2}\sin ^{2}\theta }}-mg\ell \cos \theta .} Hamilton's equations give

3696-1030: The Ising model both on and off criticality. Given any subset of spins σ A {\displaystyle \sigma _{A}} and σ B {\displaystyle \sigma _{B}} on the lattice, the following inequality holds, ⟨ σ A σ B ⟩ ≥ ⟨ σ A ⟩ ⟨ σ B ⟩ , {\displaystyle \langle \sigma _{A}\sigma _{B}\rangle \geq \langle \sigma _{A}\rangle \langle \sigma _{B}\rangle ,} where ⟨ σ A ⟩ = ⟨ ∏ j ∈ A σ j ⟩ {\displaystyle \langle \sigma _{A}\rangle =\langle \prod _{j\in A}\sigma _{j}\rangle } . With B = ∅ {\displaystyle B=\emptyset } ,

3780-4224: The Lagrangian into the result gives H = ∑ i = 1 n ( ∂ ( T ( q , q ˙ , t ) − V ( q , q ˙ , t ) ) ∂ q ˙ i q ˙ i ) − ( T ( q , q ˙ , t ) − V ( q , q ˙ , t ) ) = ∑ i = 1 n ( ∂ T ( q , q ˙ , t ) ∂ q ˙ i q ˙ i − ∂ V ( q , q ˙ , t ) ∂ q ˙ i q ˙ i ) − T ( q , q ˙ , t ) + V ( q , q ˙ , t ) {\displaystyle {\begin{aligned}{\mathcal {H}}&=\sum _{i=1}^{n}\left({\frac {\partial \left(T({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)-V({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)\right)}{\partial {\dot {q}}_{i}}}{\dot {q}}_{i}\right)-\left(T({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)-V({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)\right)\\&=\sum _{i=1}^{n}\left({\frac {\partial T({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)}{\partial {\dot {q}}_{i}}}{\dot {q}}_{i}-{\frac {\partial V({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)}{\partial {\dot {q}}_{i}}}{\dot {q}}_{i}\right)-T({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)+V({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)\end{aligned}}} Now assume that ∂ V ( q , q ˙ , t ) ∂ q ˙ i = 0 , ∀ i {\displaystyle {\frac {\partial V({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)}{\partial {\dot {q}}_{i}}}=0\;,\quad \forall i} and also assume that ∂ T ( q , q ˙ , t ) ∂ t = 0 {\displaystyle {\frac {\partial T({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)}{\partial t}}=0} Applying these assumptions results in H = ∑ i = 1 n ( ∂ T ( q , q ˙ ) ∂ q ˙ i q ˙ i − ∂ V ( q , t ) ∂ q ˙ i q ˙ i ) − T ( q , q ˙ ) + V ( q , t ) = ∑ i = 1 n ( ∂ T ( q , q ˙ ) ∂ q ˙ i q ˙ i ) − T ( q , q ˙ ) + V ( q , t ) {\displaystyle {\begin{aligned}{\mathcal {H}}&=\sum _{i=1}^{n}\left({\frac {\partial T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})}{\partial {\dot {q}}_{i}}}{\dot {q}}_{i}-{\frac {\partial V({\boldsymbol {q}},t)}{\partial {\dot {q}}_{i}}}{\dot {q}}_{i}\right)-T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})+V({\boldsymbol {q}},t)\\&=\sum _{i=1}^{n}\left({\frac {\partial T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})}{\partial {\dot {q}}_{i}}}{\dot {q}}_{i}\right)-T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})+V({\boldsymbol {q}},t)\end{aligned}}} Next assume that T

3864-1356: The Lagrangian, and then deriving the Hamiltonian from the Lagrangian. However, the relation is not true for all systems. The relation holds true for nonrelativistic systems when all of the following conditions are satisfied ∂ V ( q , q ˙ , t ) ∂ q ˙ i = 0 , ∀ i {\displaystyle {\frac {\partial V({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)}{\partial {\dot {q}}_{i}}}=0\;,\quad \forall i} ∂ T ( q , q ˙ , t ) ∂ t = 0 {\displaystyle {\frac {\partial T({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)}{\partial t}}=0} T ( q , q ˙ ) = ∑ i = 1 n ∑ j = 1 n ( c i j ( q ) q ˙ i q ˙ j ) {\displaystyle T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})=\sum _{i=1}^{n}\sum _{j=1}^{n}{\biggl (}c_{ij}({\boldsymbol {q}}){\dot {q}}_{i}{\dot {q}}_{j}{\biggr )}} where t {\displaystyle t}

3948-796: The Wikimedia System Administrators, please include the details below. Request from 172.68.168.226 via cp1108 cp1108, Varnish XID 254096987 Upstream caches: cp1108 int Error: 429, Too Many Requests at Thu, 28 Nov 2024 11:01:09 GMT Hamiltonian function 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 q ˙ i {\displaystyle {\dot {q}}^{i}} used in Lagrangian mechanics with (generalized) momenta . Both theories provide interpretations of classical mechanics and describe

4032-674: The box and y {\displaystyle y} being outside), ⟨ σ x σ y ⟩ ≤ ∑ z ∈ S ⟨ σ x σ z ⟩ ⟨ σ z σ y ⟩ . {\displaystyle \langle \sigma _{x}\sigma _{y}\rangle \leq \sum _{z\in S}\langle \sigma _{x}\sigma _{z}\rangle \langle \sigma _{z}\sigma _{y}\rangle .} Wilhelm Lenz Too Many Requests If you report this error to

4116-637: The case of time-independent H {\displaystyle {\mathcal {H}}} and ⁠ L {\displaystyle {\mathcal {L}}} ⁠ , i.e. ⁠ ∂ H / ∂ t = − ∂ L / ∂ t = 0 {\displaystyle \partial {\mathcal {H}}/\partial t=-\partial {\mathcal {L}}/\partial t=0} ⁠ , Hamilton's equations consist of 2 n first-order differential equations , while Lagrange's equations consist of n second-order equations. Hamilton's equations usually do not reduce

4200-460: The conservation of momentum also follows immediately, however all the generalized velocities q ˙ i {\displaystyle {\dot {q}}_{i}} still occur in the Lagrangian, and a system of equations in n coordinates still has to be solved. The Lagrangian and Hamiltonian approaches provide the groundwork for deeper results in classical mechanics, and suggest analogous formulations in quantum mechanics :

4284-566: The cut δ ( V + ) {\displaystyle \delta (V^{+})} to bipartite the weighted undirected graph G can be defined as | δ ( V + ) | = 1 2 ∑ i j ∈ δ ( V + ) W i j , {\displaystyle \left|\delta (V^{+})\right|={\frac {1}{2}}\sum _{ij\in \delta (V^{+})}W_{ij},} where W i j {\displaystyle W_{ij}} denotes

SECTION 50

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4368-644: The cut is the sum of the weights of the edges between S and G\S. A maximum cut size is at least the size of any other cut, varying S. For the Ising model without an external field on a graph G, the Hamiltonian becomes the following sum over the graph edges E(G) H ( σ ) = − ∑ i j ∈ E ( G ) J i j σ i σ j {\displaystyle H(\sigma )=-\sum _{ij\in E(G)}J_{ij}\sigma _{i}\sigma _{j}} . Here each vertex i of

4452-420: The difficulty of finding explicit solutions, but important theoretical results can be derived from them, because coordinates and momenta are independent variables with nearly symmetric roles. Hamilton's equations have another advantage over Lagrange's equations: if a system has a symmetry, so that some coordinate q i {\displaystyle q_{i}} does not occur in the Hamiltonian (i.e.

4536-957: The edge weight W i j = − J i j {\displaystyle W_{ij}=-J_{ij}} thus turns the Ising problem without an external field into a graph Max-Cut problem maximizing the cut size | δ ( V + ) | {\displaystyle \left|\delta (V^{+})\right|} , which is related to the Ising Hamiltonian as follows, H ( σ ) = ∑ i j ∈ E ( G ) W i j − 4 | δ ( V + ) | . {\displaystyle H(\sigma )=\sum _{ij\in E(G)}W_{ij}-4\left|\delta (V^{+})\right|.} A significant number of statistical questions to ask about this model are in

4620-2149: The equation as: d L = ∑ i ( ∂ L ∂ q i d q i + p i d q ˙ i ) + ∂ L ∂ t d t = ∑ i ( ∂ L ∂ q i d q i + d ( p i q ˙ i ) − q ˙ i d p i ) + ∂ L ∂ t d t . {\displaystyle {\begin{aligned}\mathrm {d} {\mathcal {L}}=&\sum _{i}\left({\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\,\mathrm {d} q^{i}+p_{i}\mathrm {d} {\dot {q}}^{i}\right)+{\frac {\partial {\mathcal {L}}}{\partial t}}\mathrm {d} t\\=&\sum _{i}\left({\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\,\mathrm {d} q^{i}+\mathrm {d} (p_{i}{\dot {q}}^{i})-{\dot {q}}^{i}\,\mathrm {d} p_{i}\right)+{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\,.\end{aligned}}} After rearranging, one obtains: d ( ∑ i p i q ˙ i − L ) = ∑ i ( − ∂ L ∂ q i d q i + q ˙ i d p i ) − ∂ L ∂ t d t   . {\displaystyle \mathrm {d} \!\left(\sum _{i}p_{i}{\dot {q}}^{i}-{\mathcal {L}}\right)=\sum _{i}\left(-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\,\mathrm {d} q^{i}+{\dot {q}}^{i}\mathrm {d} p_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\ .} The term in parentheses on

4704-406: The expectation (mean) value of f {\displaystyle f} . The configuration probabilities P β ( σ ) {\displaystyle P_{\beta }(\sigma )} represent the probability that (in equilibrium) the system is in a state with configuration σ {\displaystyle \sigma } . The minus sign on each term of

4788-405: The external field is zero everywhere, h  = 0, the Ising model is symmetric under switching the value of the spin in all the lattice sites; a nonzero field breaks this symmetry. Another common simplification is to assume that all of the nearest neighbors ⟨ ij ⟩ have the same interaction strength. Then we can set J ij = J for all pairs i ,  j in Λ. In this case the Hamiltonian

4872-424: The first sum is over pairs of adjacent spins (every pair is counted once). The notation ⟨ i j ⟩ {\displaystyle \langle ij\rangle } indicates that sites i {\displaystyle i} and j {\displaystyle j} are nearest neighbors. The magnetic moment is given by μ {\displaystyle \mu } . Note that

4956-702: The generalized momenta into the Hamiltonian gives H = ∑ i = 1 n ( ∂ L ( q , q ˙ , t ) ∂ q ˙ i q ˙ i ) − L ( q , q ˙ , t ) {\displaystyle {\mathcal {H}}=\sum _{i=1}^{n}\left({\frac {\partial {\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)}{\partial {\dot {q}}_{i}}}{\dot {q}}_{i}\right)-{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)} Substituting

5040-703: The graph is a spin site that takes a spin value σ i = ± 1 {\displaystyle \sigma _{i}=\pm 1} . A given spin configuration σ {\displaystyle \sigma } partitions the set of vertices V ( G ) {\displaystyle V(G)} into two σ {\displaystyle \sigma } -depended subsets, those with spin up V + {\displaystyle V^{+}} and those with spin down V − {\displaystyle V^{-}} . We denote by δ ( V + ) {\displaystyle \delta (V^{+})}

5124-748: The kinetic energy is T ( q , q ˙ ) = 1 2 ∑ k = 1 N ( m k r ˙ k ( q , q ˙ ) ⋅ r ˙ k ( q , q ˙ ) ) {\displaystyle T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})={\frac {1}{2}}\sum _{k=1}^{N}{\biggl (}m_{k}{\dot {\mathbf {r} }}_{k}({\boldsymbol {q}},{\boldsymbol {\dot {q}}})\cdot {\dot {\mathbf {r} }}_{k}({\boldsymbol {q}},{\boldsymbol {\dot {q}}}){\biggr )}} The chain rule for many variables can be used to expand

SECTION 60

#1732791669306

5208-929: The left-hand side is just the Hamiltonian H = ∑ p i q ˙ i − L {\textstyle {\mathcal {H}}=\sum p_{i}{\dot {q}}^{i}-{\mathcal {L}}} defined previously, therefore: d H = ∑ i ( − ∂ L ∂ q i d q i + q ˙ i d p i ) − ∂ L ∂ t d t   . {\displaystyle \mathrm {d} {\mathcal {H}}=\sum _{i}\left(-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\,\mathrm {d} q^{i}+{\dot {q}}^{i}\,\mathrm {d} p_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\ .} One may also calculate

5292-462: The limit of large numbers of spins: The most studied case of the Ising model is the translation-invariant ferromagnetic zero-field model on a d -dimensional lattice, namely, Λ =  Z , J ij  = 1, h  = 0. In his 1924 PhD thesis, Ising solved the model for the d  = 1 case, which can be thought of as a linear horizontal lattice where each site only interacts with its left and right neighbor. In one dimension,

5376-656: The normalization constant Z β = ∑ σ e − β H ( σ ) {\displaystyle Z_{\beta }=\sum _{\sigma }e^{-\beta H(\sigma )}} is the partition function . For a function f {\displaystyle f} of the spins ("observable"), one denotes by ⟨ f ⟩ β = ∑ σ f ( σ ) P β ( σ ) {\displaystyle \langle f\rangle _{\beta }=\sum _{\sigma }f(\sigma )P_{\beta }(\sigma )}

5460-1151: The on-shell p i = p i ( t ) {\displaystyle p_{i}=p_{i}(t)} gives: ∂ L ∂ q i = p ˙ i   . {\displaystyle {\frac {\partial {\mathcal {L}}}{\partial q^{i}}}={\dot {p}}_{i}\ .} Thus Lagrange's equations are equivalent to Hamilton's equations: ∂ H ∂ q i = − p ˙ i , ∂ H ∂ p i = q ˙ i , ∂ H ∂ t = − ∂ L ∂ t . {\displaystyle {\frac {\partial {\mathcal {H}}}{\partial q^{i}}}=-{\dot {p}}_{i}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial p_{i}}}={\dot {q}}^{i}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial t}}=-{\frac {\partial {\mathcal {L}}}{\partial t}}\,.} In

5544-1298: The other in terms of ⁠ H {\displaystyle {\mathcal {H}}} ⁠ : ∑ i ( − ∂ L ∂ q i d q i + q ˙ i d p i ) − ∂ L ∂ t d t   =   ∑ i ( ∂ H ∂ q i d q i + ∂ H ∂ p i d p i ) + ∂ H ∂ t d t   . {\displaystyle \sum _{i}\left(-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\dot {q}}^{i}\mathrm {d} p_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\ =\ \sum _{i}\left({\frac {\partial {\mathcal {H}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\frac {\partial {\mathcal {H}}}{\partial p_{i}}}\mathrm {d} p_{i}\right)+{\frac {\partial {\mathcal {H}}}{\partial t}}\,\mathrm {d} t\ .} Since these calculations are off-shell, one can equate

5628-1282: The respective coefficients of ⁠ d q i {\displaystyle \mathrm {d} q^{i}} ⁠ , ⁠ d p i {\displaystyle \mathrm {d} p_{i}} ⁠ , ⁠ d t {\displaystyle \mathrm {d} t} ⁠ on the two sides: ∂ H ∂ q i = − ∂ L ∂ q i , ∂ H ∂ p i = q ˙ i , ∂ H ∂ t = − ∂ L ∂ t   . {\displaystyle {\frac {\partial {\mathcal {H}}}{\partial q^{i}}}=-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial p_{i}}}={\dot {q}}^{i}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial t}}=-{\partial {\mathcal {L}} \over \partial t}\ .} On-shell, one substitutes parametric functions q i = q i ( t ) {\displaystyle q^{i}=q^{i}(t)} which define

5712-491: The right hand side always evaluates to 0. To perform a change of variables inside of a partial derivative, the multivariable chain rule should be used. Hence, to avoid ambiguity, the function arguments of any term inside of a partial derivative should be stated. Additionally, this proof uses the notation f ( a , b , c ) = f ( a , b ) {\displaystyle f(a,b,c)=f(a,b)} to imply that ∂ f (

5796-515: 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 . Let ( M , L ) {\displaystyle (M,{\mathcal {L}})} be a mechanical system with configuration space M {\displaystyle M} and smooth Lagrangian L . {\displaystyle {\mathcal {L}}.} Select

5880-842: The sign in the second term of the Hamiltonian above should actually be positive because the electron's magnetic moment is antiparallel to its spin, but the negative term is used conventionally. The configuration probability is given by the Boltzmann distribution with inverse temperature β ≥ 0 {\displaystyle \beta \geq 0} : P β ( σ ) = e − β H ( σ ) Z β , {\displaystyle P_{\beta }(\sigma )={\frac {e^{-\beta H(\sigma )}}{Z_{\beta }}},} where β = 1 / ( k B T ) {\displaystyle \beta =1/(k_{\text{B}}T)} , and

5964-402: The site's spin. A spin configuration , σ = { σ k } k ∈ Λ {\displaystyle {\sigma }=\{\sigma _{k}\}_{k\in \Lambda }} is an assignment of spin value to each lattice site. For any two adjacent sites i , j ∈ Λ {\displaystyle i,j\in \Lambda } there

6048-526: The solution admits no phase transition . Namely, for any positive β, the correlations ⟨σ i σ j ⟩ decay exponentially in | i  −  j |: ⟨ σ i σ j ⟩ β ≤ C exp ⁡ ( − c ( β ) | i − j | ) , {\displaystyle \langle \sigma _{i}\sigma _{j}\rangle _{\beta }\leq C\exp \left(-c(\beta )|i-j|\right),} and

6132-434: The special case ⟨ σ A ⟩ ≥ 0 {\displaystyle \langle \sigma _{A}\rangle \geq 0} results. This means that spins are positively correlated on the Ising ferromagnet. An immediate application of this is that the magnetization of any set of spins ⟨ σ A ⟩ {\displaystyle \langle \sigma _{A}\rangle }

6216-567: The spin site wants to line up with the external field. If: Ising models are often examined without an external field interacting with the lattice, that is, h  = 0 for all j in the lattice Λ. Using this simplification, the Hamiltonian becomes H ( σ ) = − ∑ ⟨ i   j ⟩ J i j σ i σ j . {\displaystyle H(\sigma )=-\sum _{\langle i~j\rangle }J_{ij}\sigma _{i}\sigma _{j}.} When

6300-6702: The summation, evaluating the partial derivative, and rejoining the summation gives ∂ T ( q , q ˙ ) ∂ q ˙ l = ∑ i ≠ l n ∑ j ≠ l n ( c i j ( q ) ∂ [ q ˙ i q ˙ j ] ∂ q ˙ l ) + ∑ i ≠ l n ( c i l ( q ) ∂ [ q ˙ i q ˙ l ] ∂ q ˙ l ) + ∑ j ≠ l n ( c l j ( q ) ∂ [ q ˙ l q ˙ j ] ∂ q ˙ l ) + c l l ( q ) ∂ [ q ˙ l 2 ] ∂ q ˙ l = ∑ i ≠ l n ∑ j ≠ l n ( 0 ) + ∑ i ≠ l n ( c i l ( q ) q ˙ i ) + ∑ j ≠ l n ( c l j ( q ) q ˙ j ) + 2 c l l ( q ) q ˙ l = ∑ i = 1 n ( c i l ( q ) q ˙ i ) + ∑ j = 1 n ( c l j ( q ) q ˙ j ) {\displaystyle {\begin{aligned}{\frac {\partial T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})}{\partial {\dot {q}}_{l}}}&=\sum _{i\neq l}^{n}\sum _{j\neq l}^{n}{\biggl (}c_{ij}({\boldsymbol {q}}){\frac {\partial \left[{\dot {q}}_{i}{\dot {q}}_{j}\right]}{\partial {\dot {q}}_{l}}}{\biggr )}+\sum _{i\neq l}^{n}{\biggl (}c_{il}({\boldsymbol {q}}){\frac {\partial \left[{\dot {q}}_{i}{\dot {q}}_{l}\right]}{\partial {\dot {q}}_{l}}}{\biggr )}+\sum _{j\neq l}^{n}{\biggl (}c_{lj}({\boldsymbol {q}}){\frac {\partial \left[{\dot {q}}_{l}{\dot {q}}_{j}\right]}{\partial {\dot {q}}_{l}}}{\biggr )}+c_{ll}({\boldsymbol {q}}){\frac {\partial \left[{\dot {q}}_{l}^{2}\right]}{\partial {\dot {q}}_{l}}}\\&=\sum _{i\neq l}^{n}\sum _{j\neq l}^{n}{\biggl (}0{\biggr )}+\sum _{i\neq l}^{n}{\biggl (}c_{il}({\boldsymbol {q}}){\dot {q}}_{i}{\biggr )}+\sum _{j\neq l}^{n}{\biggl (}c_{lj}({\boldsymbol {q}}){\dot {q}}_{j}{\biggr )}+2c_{ll}({\boldsymbol {q}}){\dot {q}}_{l}\\&=\sum _{i=1}^{n}{\biggl (}c_{il}({\boldsymbol {q}}){\dot {q}}_{i}{\biggr )}+\sum _{j=1}^{n}{\biggl (}c_{lj}({\boldsymbol {q}}){\dot {q}}_{j}{\biggr )}\end{aligned}}} Summing (this multiplied by q ˙ l {\displaystyle {\dot {q}}_{l}} ) over l {\displaystyle l} results in ∑ l = 1 n ( ∂ T ( q , q ˙ ) ∂ q ˙ l q ˙ l ) = ∑ l = 1 n ( ( ∑ i = 1 n ( c i l ( q ) q ˙ i ) + ∑ j = 1 n ( c l j ( q ) q ˙ j ) ) q ˙ l ) = ∑ l = 1 n ∑ i = 1 n ( c i l ( q ) q ˙ i q ˙ l ) + ∑ l = 1 n ∑ j = 1 n ( c l j ( q ) q ˙ j q ˙ l ) = ∑ i = 1 n ∑ l = 1 n ( c i l ( q ) q ˙ i q ˙ l ) + ∑ l = 1 n ∑ j = 1 n ( c l j ( q ) q ˙ l q ˙ j ) = T ( q , q ˙ ) + T ( q , q ˙ ) = 2 T ( q , q ˙ ) {\displaystyle {\begin{aligned}\sum _{l=1}^{n}\left({\frac {\partial T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})}{\partial {\dot {q}}_{l}}}{\dot {q}}_{l}\right)&=\sum _{l=1}^{n}\left(\left(\sum _{i=1}^{n}{\biggl (}c_{il}({\boldsymbol {q}}){\dot {q}}_{i}{\biggr )}+\sum _{j=1}^{n}{\biggl (}c_{lj}({\boldsymbol {q}}){\dot {q}}_{j}{\biggr )}\right){\dot {q}}_{l}\right)\\&=\sum _{l=1}^{n}\sum _{i=1}^{n}{\biggl (}c_{il}({\boldsymbol {q}}){\dot {q}}_{i}{\dot {q}}_{l}{\biggr )}+\sum _{l=1}^{n}\sum _{j=1}^{n}{\biggl (}c_{lj}({\boldsymbol {q}}){\dot {q}}_{j}{\dot {q}}_{l}{\biggr )}\\&=\sum _{i=1}^{n}\sum _{l=1}^{n}{\biggl (}c_{il}({\boldsymbol {q}}){\dot {q}}_{i}{\dot {q}}_{l}{\biggr )}+\sum _{l=1}^{n}\sum _{j=1}^{n}{\biggl (}c_{lj}({\boldsymbol {q}}){\dot {q}}_{l}{\dot {q}}_{j}{\biggr )}\\&=T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})+T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})\\&=2T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})\end{aligned}}} This simplification

6384-1119: The system around the vertical axis. Being absent from the Hamiltonian, azimuth φ {\displaystyle \varphi } is a cyclic coordinate , which implies conservation of its conjugate momentum. Hamilton's equations can be derived by a calculation with the Lagrangian ⁠ L {\displaystyle {\mathcal {L}}} ⁠ , generalized positions q , and generalized velocities ⋅ q , where ⁠ i = 1 , … , n {\displaystyle i=1,\ldots ,n} ⁠ . Here we work off-shell , meaning ⁠ q i {\displaystyle q^{i}} ⁠ , ⁠ q ˙ i {\displaystyle {\dot {q}}^{i}} ⁠ , ⁠ t {\displaystyle t} ⁠ are independent coordinates in phase space, not constrained to follow any equations of motion (in particular, q ˙ i {\displaystyle {\dot {q}}^{i}}

6468-644: The system is disordered. On the basis of this result, he incorrectly concluded that this model does not exhibit phase behaviour in any dimension. The Ising model undergoes a phase transition between an ordered and a disordered phase in 2 dimensions or more. Namely, the system is disordered for small β, whereas for large β the system exhibits ferromagnetic order: ⟨ σ i σ j ⟩ β ≥ c ( β ) > 0. {\displaystyle \langle \sigma _{i}\sigma _{j}\rangle _{\beta }\geq c(\beta )>0.} This

6552-459: The time derivative of q is the velocity, and so the first Hamilton equation means that the particle's velocity equals the derivative of its kinetic energy with respect to its momentum. The time derivative of the momentum p equals the Newtonian force , and so the second Hamilton equation means that the force equals the negative gradient of potential energy. A spherical pendulum consists of

6636-1260: The time evolution of coordinates and conjugate momenta in four first-order differential equations, θ ˙ = P θ m ℓ 2 φ ˙ = P φ m ℓ 2 sin 2 ⁡ θ P θ ˙ = P φ 2 m ℓ 2 sin 3 ⁡ θ cos ⁡ θ − m g ℓ sin ⁡ θ P φ ˙ = 0. {\displaystyle {\begin{aligned}{\dot {\theta }}&={P_{\theta } \over m\ell ^{2}}\\[6pt]{\dot {\varphi }}&={P_{\varphi } \over m\ell ^{2}\sin ^{2}\theta }\\[6pt]{\dot {P_{\theta }}}&={P_{\varphi }^{2} \over m\ell ^{2}\sin ^{3}\theta }\cos \theta -mg\ell \sin \theta \\[6pt]{\dot {P_{\varphi }}}&=0.\end{aligned}}} Momentum ⁠ P φ {\displaystyle P_{\varphi }} ⁠ , which corresponds to

6720-1530: The total differential of the Hamiltonian H {\displaystyle {\mathcal {H}}} with respect to coordinates ⁠ q i {\displaystyle q^{i}} ⁠ , ⁠ p i {\displaystyle p_{i}} ⁠ , ⁠ t {\displaystyle t} ⁠ instead of ⁠ q i {\displaystyle q^{i}} ⁠ , ⁠ q ˙ i {\displaystyle {\dot {q}}^{i}} ⁠ , ⁠ t {\displaystyle t} ⁠ , yielding: d H = ∑ i ( ∂ H ∂ q i d q i + ∂ H ∂ p i d p i ) + ∂ H ∂ t d t   . {\displaystyle \mathrm {d} {\mathcal {H}}=\sum _{i}\left({\frac {\partial {\mathcal {H}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\frac {\partial {\mathcal {H}}}{\partial p_{i}}}\mathrm {d} p_{i}\right)+{\frac {\partial {\mathcal {H}}}{\partial t}}\,\mathrm {d} t\ .} One may now equate these two expressions for ⁠ d H {\displaystyle d{\mathcal {H}}} ⁠ , one in terms of ⁠ L {\displaystyle {\mathcal {L}}} ⁠ ,

6804-494: The total sum in the first term does not depend on σ {\displaystyle \sigma } , imply that minimizing H ( σ ) {\displaystyle H(\sigma )} in σ {\displaystyle \sigma } is equivalent to minimizing ∑ i j ∈ δ ( V + ) J i j {\displaystyle \sum _{ij\in \delta (V^{+})}J_{ij}} . Defining

6888-616: The velocities q ˙ = ( q ˙ 1 , … , q ˙ n ) {\displaystyle {\boldsymbol {\dot {q}}}=({\dot {q}}^{1},\ldots ,{\dot {q}}^{n})} are found from the ( n {\displaystyle n} -dimensional) equation p = ∂ L / ∂ q ˙ {\displaystyle \textstyle {\boldsymbol {p}}={\partial {\mathcal {L}}}/{\partial {\boldsymbol {\dot {q}}}}} which, by assumption,

6972-418: The vertical component of angular momentum ⁠ L z = ℓ sin ⁡ θ × m ℓ sin ⁡ θ φ ˙ {\displaystyle L_{z}=\ell \sin \theta \times m\ell \sin \theta \,{\dot {\varphi }}} ⁠ , is a constant of motion. That is a consequence of the rotational symmetry of

7056-401: Was first proven by Rudolf Peierls in 1936, using what is now called a Peierls argument . The Ising model on a two-dimensional square lattice with no magnetic field was analytically solved by Lars Onsager  ( 1944 ). Onsager showed that the correlation functions and free energy of the Ising model are determined by a noninteracting lattice fermion. Onsager announced the formula for

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