Hamiltonian vector field

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In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations in classical mechanics. The integral curves of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The diffeomorphisms of a symplectic manifold arising from the flow of a Hamiltonian vector field are known as canonical transformations in physics and (Hamiltonian) symplectomorphisms in mathematics.[1]

Hamiltonian vector fields can be defined more generally on an arbitrary Poisson manifold. The Lie bracket of two Hamiltonian vector fields corresponding to functions f {\displaystyle f} {\displaystyle f} and g {\displaystyle g} {\displaystyle g} on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by the Poisson bracket of f {\displaystyle f} {\displaystyle f} and g {\displaystyle g} {\displaystyle g}.

Definition

Suppose that ( M , ω ) {\displaystyle (M,\omega )} {\displaystyle (M,\omega )} is a symplectic manifold. Since the symplectic form ω {\displaystyle \omega } {\displaystyle \omega } is nondegenerate, it sets up a fiberwise-linear isomorphism

ω : T M → T ∗ M , {\displaystyle \omega :TM\to T^{*}M,} {\displaystyle \omega :TM\to T^{*}M,}

between the tangent bundle T M {\displaystyle TM} {\displaystyle TM} and the cotangent bundle T ∗ M {\displaystyle T^{*}M} {\displaystyle T^{*}M}, with the inverse

Ω : T ∗ M → T M , Ω = ω − 1 . {\displaystyle \Omega :T^{*}M\to TM,\quad \Omega =\omega ^{-1}.} {\displaystyle \Omega :T^{*}M\to TM,\quad \Omega =\omega ^{-1}.}

Therefore, one-forms on a symplectic manifold M {\displaystyle M} {\displaystyle M} may be identified with vector fields and every differentiable function H : M → R {\displaystyle H:M\rightarrow \mathbb {R} } {\displaystyle H:M\rightarrow \mathbb {R} } determines a unique vector field X H {\displaystyle X_{H}} {\displaystyle X_{H}}, called the Hamiltonian vector field with the Hamiltonian H {\displaystyle H} {\displaystyle H}, by defining for every vector field Y {\displaystyle Y} {\displaystyle Y} on M {\displaystyle M} {\displaystyle M},

d H ( Y ) = ω ( X H , Y ) . {\displaystyle \mathrm {d} H(Y)=\omega (X_{H},Y).} {\displaystyle \mathrm {d} H(Y)=\omega (X_{H},Y).}Or more succinctly, ι X H ω = d H {\displaystyle \iota _{X_{H}}\omega =dH} {\displaystyle \iota _{X_{H}}\omega =dH}.

Note: Some authors define the Hamiltonian vector field with the opposite sign. One has to be mindful of varying conventions in physical and mathematical literature.

Examples

Suppose that M {\displaystyle M} {\displaystyle M} is a 2 n {\displaystyle 2n} {\displaystyle 2n}-dimensional symplectic manifold. Then locally, one may choose canonical coordinates ( q 1 , ⋯ , q n , p 1 , ⋯ , p n ) {\displaystyle (q^{1},\cdots ,q^{n},p_{1},\cdots ,p_{n})} {\displaystyle (q^{1},\cdots ,q^{n},p_{1},\cdots ,p_{n})} on M {\displaystyle M} {\displaystyle M}, in which the symplectic form is expressed as:[2] ω = ∑ i d q i ∧ d p i , {\displaystyle \omega =\sum _{i}\mathrm {d} q^{i}\wedge \mathrm {d} p_{i},} {\displaystyle \omega =\sum _{i}\mathrm {d} q^{i}\wedge \mathrm {d} p_{i},}

where d {\displaystyle \operatorname {d} } {\displaystyle \operatorname {d} } denotes the exterior derivative and ∧ {\displaystyle \wedge } {\displaystyle \wedge } denotes the exterior product. Then the Hamiltonian vector field with Hamiltonian H {\displaystyle H} {\displaystyle H} takes the form:[1] X H = ( ∂ H ∂ p i , − ∂ H ∂ q i ) = Ω d H , {\displaystyle \mathrm {X} _{H}=\left({\frac {\partial H}{\partial p_{i}}},-{\frac {\partial H}{\partial q^{i}}}\right)=\Omega \,\mathrm {d} H,} {\displaystyle \mathrm {X} _{H}=\left({\frac {\partial H}{\partial p_{i}}},-{\frac {\partial H}{\partial q^{i}}}\right)=\Omega \,\mathrm {d} H,}

where Ω {\displaystyle \Omega } {\displaystyle \Omega } is a 2 n × 2 n {\displaystyle 2n\times 2n} {\displaystyle 2n\times 2n} square matrix

Ω = [ 0 I n − I n 0 ] , {\displaystyle \Omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}},} {\displaystyle \Omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}},}

and

d H = [ ∂ H ∂ q i ∂ H ∂ p i ] . {\displaystyle \mathrm {d} H={\begin{bmatrix}{\frac {\partial H}{\partial q^{i}}}\\{\frac {\partial H}{\partial p_{i}}}\end{bmatrix}}.} {\displaystyle \mathrm {d} H={\begin{bmatrix}{\frac {\partial H}{\partial q^{i}}}\\{\frac {\partial H}{\partial p_{i}}}\end{bmatrix}}.}

The matrix Ω {\displaystyle \Omega } {\displaystyle \Omega } is frequently denoted with J {\displaystyle \mathbf {J} } {\displaystyle \mathbf {J} }.

Suppose that M = R 2 n {\displaystyle M=\mathbb {R} ^{2n}} {\displaystyle M=\mathbb {R} ^{2n}} is the 2 n {\displaystyle 2n} {\displaystyle 2n}-dimensional symplectic vector space with (global) canonical coordinates.

  • If H = p i {\displaystyle H=p_{i}} {\displaystyle H=p_{i}} then X H = ∂ / ∂ q i ; {\displaystyle X_{H}=\partial /\partial q^{i};} {\displaystyle X_{H}=\partial /\partial q^{i};}
  • if H = q i {\displaystyle H=q_{i}} {\displaystyle H=q_{i}} then X H = − ∂ / ∂ p i ; {\displaystyle X_{H}=-\partial /\partial p^{i};} {\displaystyle X_{H}=-\partial /\partial p^{i};}
  • if H = 1 2 ∑ ( p i ) 2 {\textstyle H={\frac {1}{2}}\sum (p_{i})^{2}} {\textstyle H={\frac {1}{2}}\sum (p_{i})^{2}} then X H = ∑ p i ∂ / ∂ q i ; {\textstyle X_{H}=\sum p_{i}\partial /\partial q^{i};} {\textstyle X_{H}=\sum p_{i}\partial /\partial q^{i};}
  • if H = 1 2 ∑ a i j q i q j , a i j = a j i {\textstyle H={\frac {1}{2}}\sum a_{ij}q^{i}q^{j},a_{ij}=a_{ji}} {\textstyle H={\frac {1}{2}}\sum a_{ij}q^{i}q^{j},a_{ij}=a_{ji}} then X H = − ∑ a i j q i ∂ / ∂ p j . {\textstyle X_{H}=-\sum a_{ij}q_{i}\partial /\partial p^{j}.} {\textstyle X_{H}=-\sum a_{ij}q_{i}\partial /\partial p^{j}.}

Properties

  • The assignment f ↦ X f {\displaystyle f\mapsto X_{f}} {\displaystyle f\mapsto X_{f}} is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.
  • Suppose that ( q 1 , ⋯ , q n , p 1 , ⋯ , p n ) {\displaystyle (q^{1},\cdots ,q^{n},p_{1},\cdots ,p_{n})} {\displaystyle (q^{1},\cdots ,q^{n},p_{1},\cdots ,p_{n})} are canonical coordinates on M {\displaystyle M} {\displaystyle M} (see above). Then a curve γ ( t ) = ( q ( t ) , p ( t ) ) {\displaystyle \gamma (t)=(q(t),p(t))} {\displaystyle \gamma (t)=(q(t),p(t))} is an integral curve of the Hamiltonian vector field X H {\displaystyle X_{H}} {\displaystyle X_{H}} if and only if it is a solution of Hamilton's equations:[1] q ˙ i = ∂ H ∂ p i p ˙ i = − ∂ H ∂ q i . {\displaystyle {\begin{aligned}{\dot {q}}^{i}&={\frac {\partial H}{\partial p_{i}}}\\{\dot {p}}_{i}&=-{\frac {\partial H}{\partial q^{i}}}.\end{aligned}}} {\displaystyle {\begin{aligned}{\dot {q}}^{i}&={\frac {\partial H}{\partial p_{i}}}\\{\dot {p}}_{i}&=-{\frac {\partial H}{\partial q^{i}}}.\end{aligned}}}
  • The Hamiltonian H {\displaystyle H} {\displaystyle H} is constant along the integral curves, because ⟨ d H , γ ˙ ⟩ = ω ( X H ( γ ) , X H ( γ ) ) = 0 {\displaystyle \langle dH,{\dot {\gamma }}\rangle =\omega (X_{H}(\gamma ),X_{H}(\gamma ))=0} {\displaystyle \langle dH,{\dot {\gamma }}\rangle =\omega (X_{H}(\gamma ),X_{H}(\gamma ))=0}. That is, H ( γ ( t ) ) {\displaystyle H(\gamma (t))} {\displaystyle H(\gamma (t))} is actually independent of t {\displaystyle t} {\displaystyle t}. This property corresponds to the conservation of energy in Hamiltonian mechanics.
  • More generally, if two functions F {\displaystyle F} {\displaystyle F} and H {\displaystyle H} {\displaystyle H} have a zero Poisson bracket (cf. below), then F {\displaystyle F} {\displaystyle F} is constant along the integral curves of H {\displaystyle H} {\displaystyle H}, and similarly, H {\displaystyle H} {\displaystyle H} is constant along the integral curves of F {\displaystyle F} {\displaystyle F}. This fact is the abstract mathematical principle behind Noether's theorem.[nb 1]
  • The symplectic form ω {\displaystyle \omega } {\displaystyle \omega } is preserved by the Hamiltonian flow. Equivalently, the Lie derivative L X H ω = 0 {\displaystyle {\mathcal {L}}_{X_{H}}\omega =0} {\displaystyle {\mathcal {L}}_{X_{H}}\omega =0}.

Poisson bracket

The notion of a Hamiltonian vector field leads to a skew-symmetric bilinear operation on the differentiable functions on a symplectic manifold M {\displaystyle M} {\displaystyle M}, the Poisson bracket, defined by the formula

{ f , g } = ω ( X g , X f ) = d g ( X f ) = L X f g {\displaystyle \{f,g\}=\omega (X_{g},X_{f})=dg(X_{f})={\mathcal {L}}_{X_{f}}g} {\displaystyle \{f,g\}=\omega (X_{g},X_{f})=dg(X_{f})={\mathcal {L}}_{X_{f}}g}

where L X {\displaystyle {\mathcal {L}}_{X}} {\displaystyle {\mathcal {L}}_{X}} denotes the Lie derivative along a vector field X {\displaystyle X} {\displaystyle X}. Moreover, one can check that the following identity holds:[1] X { f , g } = − [ X f , X g ] {\displaystyle X_{\{f,g\}}=-[X_{f},X_{g}]} {\displaystyle X_{\{f,g\}}=-[X_{f},X_{g}]},

where the right hand side represents the Lie bracket of the Hamiltonian vector fields with Hamiltonians f {\displaystyle f} {\displaystyle f} and g {\displaystyle g} {\displaystyle g}. As a consequence (a proof at Poisson bracket), the Poisson bracket satisfies the Jacobi identity:[1] { { f , g } , h } + { { g , h } , f } + { { h , f } , g } = 0 {\displaystyle \{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0} {\displaystyle \{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0},

which means that the vector space of differentiable functions on M {\displaystyle M} {\displaystyle M}, endowed with the Poisson bracket, has the structure of a Lie algebra over R {\displaystyle \mathbb {R} } {\displaystyle \mathbb {R} }, and the assignment f ↦ X f {\displaystyle f\mapsto X_{f}} {\displaystyle f\mapsto X_{f}} is a Lie algebra homomorphism, whose kernel consists of the locally constant functions (constant functions if M {\displaystyle M} {\displaystyle M} is connected).

Remarks

  1. See Lee (2003, Chapter 18) for a very concise statement and proof of Noether's theorem.

Notes

  1. Lee 2003, Chapter 18.
  2. Lee 2003, Chapter 12.

Works cited