Hamiltonian system

Informally, a Hamiltonian system is a mathematical formalism developed by Hamilton to describe the evolution equations of a physical system. The advantage of this description is that it gives important insights into the dynamics, even if the initial value problem cannot be solved analytically. One example is the planetary movement of three bodies: while there is no closed-form solution to the general problem, Poincaré showed for the first time that it exhibits deterministic chaos.

Formally, a Hamiltonian system is a dynamical system characterised by the scalar function ${\displaystyle H({\boldsymbol {q}},{\boldsymbol {p}},t)}$, also known as the Hamiltonian.[1] The state of the system, ${\displaystyle {\boldsymbol {r}}}$, is described by the generalized coordinates ${\displaystyle {\boldsymbol {p}}}$ and ${\displaystyle {\boldsymbol {q}}}$, corresponding to generalized momentum and position respectively. Both ${\displaystyle {\boldsymbol {p}}}$ and ${\displaystyle {\boldsymbol {q}}}$ are real-valued vectors with the same dimension N. Thus, the state is completely described by the 2N-dimensional vector

${\displaystyle {\boldsymbol {r}}=({\boldsymbol {q}},{\boldsymbol {p}})}$

and the evolution equations are given by Hamilton's equations:

${\displaystyle {\begin{aligned}&{\frac {d{\boldsymbol {p}}}{dt}}=-{\frac {\partial H}{\partial {\boldsymbol {q}}}},\\[5pt]&{\frac {d{\boldsymbol {q}}}{dt}}=+{\frac {\partial H}{\partial {\boldsymbol {p}}}}.\end{aligned}}}$

The trajectory ${\displaystyle {\boldsymbol {r}}(t)}$ is the solution of the initial value problem defined by Hamilton's equations and the initial condition ${\displaystyle {\boldsymbol {r}}(t=0)={\boldsymbol {r}}_{0}\in \mathbb {R} ^{2N}}$.

If the Hamiltonian is not explicitly time-dependent, i.e. if ${\displaystyle H({\boldsymbol {q}},{\boldsymbol {p}},t)=H({\boldsymbol {q}},{\boldsymbol {p}})}$, then the Hamiltonian does not vary with time at all:[1]

and thus the Hamiltonian is a constant of motion, whose constant equals the total energy of the system: ${\displaystyle H=E}$. Examples of such systems are the undamped pendulum, the harmonic oscillator, and dynamical billiards.

One important property of a Hamiltonian dynamical system is that it has a symplectic structure.[1] Writing

${\displaystyle \nabla _{\boldsymbol {r}}H({\boldsymbol {r}})={\begin{bmatrix}{\frac {\partial H({\boldsymbol {q}},{\boldsymbol {p}})}{\partial {\boldsymbol {q}}}}\\{\frac {\partial H({\boldsymbol {q}},{\boldsymbol {p}})}{\partial {\boldsymbol {p}}}}\\\end{bmatrix}}}$

the evolution equation of the dynamical system can be written as

${\displaystyle {\frac {d{\boldsymbol {r}}}{dt}}=M_{N}\nabla _{\boldsymbol {r}}H({\boldsymbol {r}})}$

where

${\displaystyle M_{N}={\begin{bmatrix}0&I_{N}\\-I_{N}&0\\\end{bmatrix}}}$

and I_N is the N×N identity matrix.

One important consequence of this property is that an infinitesimal phase-space volume is preserved.[1] A corollary of this is Liouville's theorem, which states that on a Hamiltonian system, the phase-space volume of a closed surface is preserved under time evolution.[1]

${\displaystyle {\begin{aligned}{\frac {d}{dt}}\oint _{\partial V}d{\boldsymbol {r}}&=\oint _{\partial V}{\frac {d{\boldsymbol {r}}}{dt}}\cdot d{\hat {\boldsymbol {n}}}_{\partial V}\\&=\oint _{\partial V}\left(M_{N}\nabla _{\boldsymbol {r}}H({\boldsymbol {r}})\right)\cdot d{\hat {\boldsymbol {n}}}_{\partial V}\\&=\int _{V}\nabla _{\boldsymbol {r}}\cdot \left(M_{N}\nabla _{\boldsymbol {r}}H({\boldsymbol {r}})\right)\,dV\\&=\int _{V}\sum _{i=1}^{N}\sum _{j=1}^{N}\left({\frac {\partial ^{2}H}{\partial q_{i}\partial p_{j}}}-{\frac {\partial ^{2}H}{\partial p_{i}\partial q_{j}}}\right)\,dV\\&=0\end{aligned}}}$

where the third equality comes from the divergence theorem.

Certain Hamiltonian systems exhibit chaotic behavior. When the evolution of a Hamiltonian system is highly sensitive to initial conditions, and the motion appears random and erratic, the system is said to exhibit Hamiltonian chaos.

• Dynamical billiards
• Planetary systems, more specifically, the n-body problem.
• Canonical general relativity

• Action-angle coordinates
• Liouville's theorem
• Integrable system
• Symplectic manifold
• Kolmogorov–Arnold–Moser theorem

• Poincaré recurrence theorem
• Lyapunov exponent
• Three-body problem
• Ergodic theory

• Almeida, A. M. (1992). Hamiltonian systems: Chaos and quantization. Cambridge monographs on mathematical physics. Cambridge (u.a.: Cambridge Univ. Press)
• Audin, M., (2008). Hamiltonian systems and their integrability. Providence, R.I: American Mathematical Society, ISBN 978-0-8218-4413-7
• Dickey, L. A. (2003). Soliton equations and Hamiltonian systems. Advanced series in mathematical physics, v. 26. River Edge, NJ: World Scientific.
• Treschev, D., & Zubelevich, O. (2010). Introduction to the perturbation theory of Hamiltonian systems. Heidelberg: Springer
• Zaslavsky, G. M. (2007). The physics of chaos in Hamiltonian systems. London: Imperial College Press.

• James Meiss (ed.). "Hamiltonian Systems". Scholarpedia.