Misplaced Pages

#191808

105-502: Liouville's theorem applies to conservative systems , that is, systems in which the effects of friction are absent or can be ignored. The general mathematical formulation for such systems is the measure-preserving dynamical system . Liouville's theorem applies when there are degrees of freedom that can be interpreted as positions and momenta; not all measure-preserving dynamical systems have these, but Hamiltonian systems do. The general setting for conjugate position and momentum coordinates

210-911: 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 (

315-520: 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

420-671: 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 (

525-450: 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 = ∫

630-1651: 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

735-447: A Hamiltonian dynamical system with canonical coordinates q i {\displaystyle q_{i}} and conjugate momenta p i {\displaystyle p_{i}} , where i = 1 , … , n {\displaystyle i=1,\dots ,n} . Then the phase space distribution ρ ( p , q ) {\displaystyle \rho (p,q)} determines

840-469: 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,

945-799: 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

1050-433: A classical system using Hamiltonian mechanics. Classical variables are then re-interpreted as quantum operators, while Poisson brackets are replaced by commutators . In this case, the resulting equation is where ρ is the density matrix . When applied to the expectation value of an observable , the corresponding equation is given by Ehrenfest's theorem , and takes the form where A {\displaystyle A}

1155-1190: 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

SECTION 10

#1732856206192

1260-414: A given system, we can consider the phase space ( q μ , p μ ) {\displaystyle (q^{\mu },p_{\mu })} of a particular Hamiltonian H {\displaystyle H} as a manifold ( M , ω ) {\displaystyle (M,\omega )} endowed with a symplectic 2-form The volume form of our manifold

1365-414: A non-singular transformation τ : X → X {\displaystyle \tau :X\to X} , the following statements are equivalent: The above implies that, if μ ( X ) < ∞ {\displaystyle \mu (X)<\infty } and τ {\displaystyle \tau } is measure-preserving, then the dynamical system

1470-477: A smooth measure (locally, this measure is the 6 n -dimensional Lebesgue measure ). The theorem says this smooth measure is invariant under the Hamiltonian flow . More generally, one can describe the necessary and sufficient condition under which a smooth measure is invariant under a flow. The Hamiltonian case then becomes a corollary. We can also formulate Liouville's Theorem in terms of symplectic geometry . For

1575-757: 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

1680-642: A system of multiple particles, each one will have a phase-space trajectory that traces out an ellipse corresponding to the particle's energy. The frequency at which the ellipse is traced is given by the ω {\displaystyle \omega } in the Hamiltonian, independent of any differences in energy. As a result, a region of phase space will simply rotate about the point ( q , p ) = ( 0 , 0 ) {\displaystyle (\mathbf {q} ,\mathbf {p} )=(0,0)} with frequency dependent on ω {\displaystyle \omega } . This can be seen in

1785-443: 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

1890-488: 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

1995-754: 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

2100-420: A very similar procedure to the undamped harmonic oscillator case, and we arrive again at Plugging in our modified Hamilton's equations, we find Calculating our new infinitesimal phase space volume, and keeping only first order in δ t {\displaystyle \delta t} we find the following result: We have found that the infinitesimal phase-space volume is no longer constant, and thus

2205-418: A wandering (dissipative) set. A commonplace informal example of Hopf decomposition is the mixing of two liquids (some textbooks mention rum and coke): The initial state, where the two liquids are not yet mixed, can never recur again after mixing; it is part of the dissipative set. Likewise any of the partially-mixed states. The result, after mixing (a cuba libre , in the canonical example), is stable, and forms

SECTION 20

#1732856206192

2310-506: A wandering set of τ {\displaystyle \tau } . By definition of wandering sets and since τ {\displaystyle \tau } preserves μ {\displaystyle \mu } , X {\displaystyle X} would thus contain a countably infinite union of pairwise disjoint sets that have the same μ {\displaystyle \mu } -measure as A {\displaystyle A} . Since it

2415-461: A well-defined past state. A measurable transformation τ : X → X {\displaystyle \tau :X\to X} is called non-singular when μ ( τ − 1 σ ) = 0 {\displaystyle \mu (\tau ^{-1}\sigma )=0} if and only if μ ( σ ) = 0 {\displaystyle \mu (\sigma )=0} . In this case,

2520-680: 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

2625-430: Is a Borel space ( X , Σ) equipped with a sigma-finite measure μ and a transformation τ . Here, X is a set , and Σ is a sigma-algebra on X , so that the pair ( X , Σ) is a measurable space . μ is a sigma-finite measure on the sigma-algebra. The space X is the phase space of the dynamical system. A transformation (a map) τ : X → X {\displaystyle \tau :X\to X}

2730-631: Is a conserved current . Notice that the difference between this and Liouville's equation are the terms where H {\displaystyle H} is the Hamiltonian, and where the derivatives ∂ q ˙ i / ∂ q i {\displaystyle \partial {\dot {q}}_{i}/\partial q_{i}} and ∂ p ˙ i / ∂ p i {\displaystyle \partial {\dot {p}}_{i}/\partial p_{i}} have been evaluated using Hamilton's equations of motion. That is, viewing

2835-948: 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,

2940-1295: 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

3045-1728: 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

3150-680: Is an observable. Note the sign difference, which follows from the assumption that the operator is stationary and the state is time-dependent. In the phase-space formulation of quantum mechanics, substituting the Moyal brackets for Poisson brackets in the phase-space analog of the von Neumann equation results in compressibility of the probability fluid , and thus violations of Liouville's theorem incompressibility. This, then, leads to concomitant difficulties in defining meaningful quantum trajectories. Consider an N {\displaystyle N} -particle system in three dimensions, and focus on only

3255-1163: 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

Liouville's theorem (Hamiltonian) - Misplaced Pages Continue

3360-416: Is available in the mathematical setting of symplectic geometry . Liouville's theorem ignores the possibility of chemical reactions , where the total number of particles may change over time, or where energy may be transferred to internal degrees of freedom . There are extensions of Liouville's theorem to cover these various generalized settings, including stochastic systems. The Liouville equation describes

3465-709: Is called invariant , or, more commonly, a measure-preserving dynamical system . A non-singular dynamical system is conservative if, for every set σ ∈ Σ {\displaystyle \sigma \in \Sigma } of positive measure and for every n ∈ N {\displaystyle n\in \mathbb {N} } , one has some integer p > n {\displaystyle p>n} such that μ ( σ ∩ τ − p σ ) > 0 {\displaystyle \mu (\sigma \cap \tau ^{-p}\sigma )>0} . Informally, this can be interpreted as saying that

3570-567: Is conservative. This is effectively the modern statement of the Poincaré recurrence theorem . A sketch of a proof of the equivalence of these four properties is given in the article on the Hopf decomposition . Suppose that μ ( X ) < ∞ {\displaystyle \mu (X)<\infty } and τ {\displaystyle \tau } is measure-preserving. Let A {\displaystyle A} be

3675-530: Is ergodic, the following statements are equivalent: Hamilton%27s 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 q ˙ i {\displaystyle {\dot {q}}^{i}} used in Lagrangian mechanics with (generalized) momenta . Both theories provide interpretations of classical mechanics and describe

3780-402: Is given by the ellipse of constant H {\displaystyle H} . Explicitly, one can solve Hamilton's equations for the system and find where Q i {\displaystyle Q_{i}} and P i {\displaystyle P_{i}} denote the initial position and momentum of the i {\displaystyle i} -th particle. For

3885-426: Is given in the previous example. This time, we add the condition that each particle experiences a frictional force − γ p i {\displaystyle -\gamma p_{i}} , where γ {\displaystyle \gamma } is a positive constant dictating the amount of friction. As this is a non-conservative force , we need to extend Hamilton's equations as Unlike

3990-601: 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,

4095-531: Is individually ergodic . An informal example of this would be a tub, with a divider down the middle, with liquids filling each compartment. The liquid on one side can clearly mix with itself, and so can the other, but, due to the partition, the two sides cannot interact. Clearly, this can be treated as two independent systems; leakage between the two sides, of measure zero, can be ignored. The ergodic decomposition theorem states that all conservative systems can be split into such independent parts, and that this splitting

4200-422: Is measure-preserving, as the number of particles in the rings does not change, and, per Newtonian orbital mechanics, the phase space is incompressible: it can be stretched or squeezed, but not shrunk (this is the content of Liouville's theorem ). Formally, a measurable dynamical system is conservative if and only if it is non-singular, and has no wandering sets. A measurable dynamical system ( X , Σ, μ , τ )

4305-1174: 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

Liouville's theorem (Hamiltonian) - Misplaced Pages Continue

4410-459: Is not conservative. In fact, every interval of length strictly less than 1 {\displaystyle 1} contained in X {\displaystyle X} is wandering. In particular, X {\displaystyle X} can be written as a countable union of wandering sets. The Hopf decomposition states that every measure space with a non-singular transformation can be decomposed into an invariant conservative set and

4515-622: 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}})}

4620-453: Is often restated in terms of the Poisson bracket as or, in terms of the linear Liouville operator or Liouvillian , as In ergodic theory and dynamical systems , motivated by the physical considerations given so far, there is a corresponding result also referred to as Liouville's theorem. In Hamiltonian mechanics , the phase space is a smooth manifold that comes naturally equipped with

4725-399: Is preserved, not only its top exterior power. That is, Liouville's Theorem also gives The analog of Liouville equation in quantum mechanics describes the time evolution of a mixed state . Canonical quantization yields a quantum-mechanical version of this theorem, the von Neumann equation . This procedure, often used to devise quantum analogues of classical systems, involves describing

4830-412: Is said to be Σ-measurable if and only if, for every σ ∈ Σ, one has τ − 1 σ ∈ Σ {\displaystyle \tau ^{-1}\sigma \in \Sigma } . The transformation is a single "time-step" in the evolution of the dynamical system. One is interested in invertible transformations, so that the current state of the dynamical system came from

4935-507: Is the Bernoulli process : it is the set of all possible infinite sequences of coin flips (equivalently, the set { 0 , 1 } N {\displaystyle \{0,1\}^{\mathbb {N} }} of infinite strings of zeros and ones); each individual coin flip is independent of the others. The ergodic decomposition theorem states, roughly, that every conservative system can be split up into components, each component of which

5040-1394: 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}}}} ,

5145-411: 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,

5250-551: Is the top exterior power of the symplectic 2-form, and is just another representation of the measure on the phase space described above. On our phase space symplectic manifold we can define a Hamiltonian vector field generated by a function f ( q , p ) {\displaystyle f(q,p)} as Specifically, when the generating function is the Hamiltonian itself, f ( q , p ) = H {\displaystyle f(q,p)=H} , we get where we utilized Hamilton's equations of motion and

5355-565: 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

SECTION 50

#1732856206192

5460-465: Is unique (up to differences of measure zero). Thus, by convention, the study of conservative systems becomes the study of their ergodic components. Formally, every ergodic system is conservative. Recall that an invariant set σ ∈ Σ is one for which τ ( σ ) = σ . For an ergodic system, the only invariant sets are those with measure zero or with full measure (are null or are conull ); that they are conservative then follows trivially from this. When τ

5565-545: 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}})} ⁠ ,

5670-465: The n -dimensional divergence theorem . This proof is based on the fact that the evolution of ρ {\displaystyle \rho } obeys an 2n -dimensional version of the continuity equation : That is, the 3-tuple ( ρ , ρ q ˙ i , ρ p ˙ i ) {\displaystyle (\rho ,\rho {\dot {q}}_{i},\rho {\dot {p}}_{i})}

5775-574: The Lebesgue measure , and consider the shift operator τ : X → X , x ↦ x + 1 {\displaystyle \tau :X\to X,x\mapsto x+1} . Since the Lebesgue measure is translation-invariant, τ {\displaystyle \tau } is measure-preserving. However, ( X , A , μ , τ ) {\displaystyle (X,{\mathcal {A}},\mu ,\tau )}

5880-492: The measure-preserving dynamical systems . Informally, dynamical systems describe the time evolution of the phase space of some mechanical system. Commonly, such evolution is given by some differential equations, or quite often in terms of discrete time steps. However, in the present case, instead of focusing on the time evolution of discrete points, one shifts attention to the time evolution of collections of points. One such example would be Saturn's rings : rather than tracking

5985-492: 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

6090-1260: 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}})}

6195-606: 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 (

6300-493: 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

6405-1022: 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,

SECTION 60

#1732856206192

6510-1140: 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

6615-4224: 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

6720-1356: 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}

6825-417: The animation above. To see an example where Liouville's theorem does not apply, we can modify the equations of motion for the simple harmonic oscillator to account for the effects of friction or damping. Consider again the system of N {\displaystyle N} particles each in a 3 {\displaystyle 3} -dimensional isotropic harmonic potential, the Hamiltonian for which

6930-441: The case of N {\displaystyle N} 3 {\displaystyle 3} -dimensional isotropic harmonic oscillators. That is, each particle in our ensemble can be treated as a simple harmonic oscillator . The Hamiltonian for this system is given by By using Hamilton's equations with the above Hamiltonian we find that the term in parentheses above is identically zero, thus yielding From this we can find

7035-637: 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

7140-460: 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 :

7245-482: The conservative set; further mixing does not alter it. In this example, the conservative set is also ergodic: if one added one more drop of liquid (say, lemon juice), it would not stay in one place, but would come to mix in everywhere. One word of caution about this example: although mixing systems are ergodic, ergodic systems are not in general mixing systems! Mixing implies an interaction which may not exist. The canonical example of an ergodic system that does not mix

7350-648: The current state of the system revisits or comes arbitrarily close to a prior state; see Poincaré recurrence for more. A non-singular transformation τ : X → X {\displaystyle \tau :X\to X} is incompressible if, whenever one has τ − 1 σ ⊂ σ {\displaystyle \tau ^{-1}\sigma \subset \sigma } , then μ ( σ ∖ τ − 1 σ ) = 0 {\displaystyle \mu (\sigma \smallsetminus \tau ^{-1}\sigma )=0} . For

7455-509: The definition of the chain rule. In this formalism, Liouville's Theorem states that the Lie derivative of the volume form is zero along the flow generated by X H {\displaystyle X_{H}} . That is, for ( M , ω ) {\displaystyle (M,\omega )} a 2n-dimensional symplectic manifold, In fact, the symplectic structure ω {\displaystyle \omega } itself

7560-420: 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.

7665-467: The dynamics, and thus, their phase space does not shrink over time. Precisely speaking, they are those dynamical systems that have a null wandering set : under time evolution, no portion of the phase space ever "wanders away", never to be returned to or revisited. Alternately, conservative systems are those to which the Poincaré recurrence theorem applies. An important special case of conservative systems are

7770-2149: 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

7875-406: The equations of motion for the simple harmonic oscillator, these modified equations do not take the form of Hamilton's equations, and therefore we do not expect Liouville's theorem to hold. Instead, as depicted in the animation in this section, a generic phase space volume will shrink as it evolves under these equations of motion. To see this violation of Liouville's theorem explicitly, we can follow

7980-456: The evolution of d N {\displaystyle \mathrm {d} {\mathcal {N}}} particles. Within phase space, these d N {\displaystyle \mathrm {d} {\mathcal {N}}} particles occupy an infinitesimal volume given by We want d N d Γ {\displaystyle {\frac {\mathrm {d} {\mathcal {N}}}{\mathrm {d} \Gamma }}} to remain

8085-467: The evolution of ρ ( p , q ; t ) {\displaystyle \rho (p,q;t)} in time t {\displaystyle t} : Time derivatives are denoted by dots, and are evaluated according to Hamilton's equations for the system. This equation demonstrates the conservation of density in phase space (which was Gibbs 's name for the theorem). Liouville's theorem states that A proof of Liouville's theorem uses

8190-702: 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

8295-447: The infinitesimal volume of phase space: Thus we have ultimately found that the infinitesimal phase-space volume is unchanged, yielding demonstrating that Liouville's theorem holds for this system. The question remains of how the phase-space volume actually evolves in time. Above we have shown that the total volume is conserved, but said nothing about what it looks like. For a single particle we can see that its trajectory in phase space

8400-450: The initial position and momentum of the i {\displaystyle i} -th particle. As the system evolves the total phase-space volume will spiral in to the origin. This can be seen in the figure above. Conservative system In mathematics , a conservative system is a dynamical system which stands in contrast to a dissipative system . Roughly speaking, such systems have no friction or other mechanism to dissipate

8505-748: 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

8610-929: 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

8715-534: The motion through phase space as a 'fluid flow' of system points, the theorem that the convective derivative of the density, d ρ / d t {\displaystyle d\rho /dt} , is zero follows from the equation of continuity by noting that the 'velocity field' ( p ˙ , q ˙ ) {\displaystyle ({\dot {p}},{\dot {q}})} in phase space has zero divergence (which follows from Hamilton's relations). The theorem above

8820-1151: 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

8925-1298: 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

9030-404: The phase-space density is not conserved. As can be seen from the equation as time increases, we expect our phase-space volume to decrease to zero as friction affects the system. As for how the phase-space volume evolves in time, we will still have the constant rotation as in the undamped case. However, the damping will introduce a steady decrease in the radii of each ellipse. Again we can solve for

9135-435: The probability ρ ( p , q ) d n q d n p {\displaystyle \rho (p,q)\;\mathrm {d} ^{n}q\,\mathrm {d} ^{n}p} that the system will be found in the infinitesimal phase space volume d n q d n p {\displaystyle \mathrm {d} ^{n}q\,\mathrm {d} ^{n}p} . The Liouville equation governs

9240-1282: 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

9345-491: 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 (

9450-515: 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

9555-1058: The same throughout time, so that ρ ( Γ , t ) {\displaystyle \rho (\Gamma ,t)} is constant along the trajectories of the system. If we allow our particles to evolve by an infinitesimal time step δ t {\displaystyle \delta t} , we see that each particle phase space location changes as where q i ˙ {\displaystyle {\dot {q_{i}}}} and p i ˙ {\displaystyle {\dot {p_{i}}}} denote d q i d t {\displaystyle {\frac {dq_{i}}{dt}}} and d p i d t {\displaystyle {\frac {dp_{i}}{dt}}} respectively, and we have only kept terms linear in δ t {\displaystyle \delta t} . Extending this to our infinitesimal hypercube d Γ {\displaystyle \mathrm {d} \Gamma } ,

9660-420: The side lengths change as To find the new infinitesimal phase-space volume d Γ ′ {\displaystyle \mathrm {d} \Gamma '} , we need the product of the above quantities. To first order in δ t {\displaystyle \delta t} , we get the following: So far, we have yet to make any specifications about our system. Let us now specialize to

9765-6702: 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

9870-597: The system ( X , Σ, μ , τ ) is called a non-singular dynamical system . The condition of being non-singular is necessary for a dynamical system to be suitable for modeling (non-equilibrium) systems. That is, if a certain configuration of the system is "impossible" (i.e. μ ( σ ) = 0 {\displaystyle \mu (\sigma )=0} ) then it must stay "impossible" (was always impossible: μ ( τ − 1 σ ) = 0 {\displaystyle \mu (\tau ^{-1}\sigma )=0} ), but otherwise,

9975-1113: 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}}

10080-715: The system can evolve arbitrarily. Non-singular systems preserve the negligible sets, but are not required to preserve any other class of sets. The sense of the word singular here is the same as in the definition of a singular measure in that no portion of μ {\displaystyle \mu } is singular with respect to μ ∘ τ − 1 {\displaystyle \mu \circ \tau ^{-1}} and vice versa. A non-singular dynamical system for which μ ( τ − 1 σ ) = μ ( σ ) {\displaystyle \mu (\tau ^{-1}\sigma )=\mu (\sigma )}

10185-454: 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

10290-1260: 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

10395-405: The time evolution of individual grains of sand in the rings, one is instead interested in the time evolution of the density of the rings: how the density thins out, spreads, or becomes concentrated. Over short time-scales (hundreds of thousands of years), Saturn's rings are stable, and are thus a reasonable example of a conservative system and more precisely, a measure-preserving dynamical system. It

10500-485: The time evolution of the phase space distribution function . Although the equation is usually referred to as the "Liouville equation", Josiah Willard Gibbs was the first to recognize the importance of this equation as the fundamental equation of statistical mechanics. It is referred to as the Liouville equation because its derivation for non-canonical systems utilises an identity first derived by Liouville in 1838. Consider

10605-1530: 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}}} ⁠ ,

10710-414: The trajectories explicitly using Hamilton's equations, taking care to use the modified ones above. Letting α ≡ γ 2 {\displaystyle \alpha \equiv {\frac {\gamma }{2}}} for convenience, we find where the values Q i {\displaystyle Q_{i}} and P i {\displaystyle P_{i}} denote

10815-616: 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,

10920-418: 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

11025-756: Was assumed μ ( X ) < ∞ {\displaystyle \mu (X)<\infty } , it follows that A {\displaystyle A} is a null set, and so all wandering sets must be null sets. This argumentation fails for even the simplest examples if μ ( X ) = ∞ {\displaystyle \mu (X)=\infty } . Indeed, consider for instance ( X , A , μ ) = ( R , B ( R ) , λ ) {\displaystyle (X,{\mathcal {A}},\mu )=(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ),\lambda )} , where λ {\displaystyle \lambda } denotes

#191808