Hamiltonian Dynamics Hamiltonians equation a dynamic system with degree of freedom d can be described by a d-dim vector q( generalized coordinate)and a d-dim vector p( generalized momentum, defined by p i ac(a, a, t) in physics), z=(q,p is called the canonical coordinates The dynamic system is dominated by its Hamiltonian: H (q, p, t)=iqipi L(a, q tlg=g(p If U does not change with time, H(a,p, t)=h(q, p), in which case H(q,p)=u+ K is the energy of the systemHamiltonian Dynamics • Hamiltonian’s Equation – A dynamic system with degree of freedom 𝑑 can be described by a 𝑑-dim vector 𝑞 (generalized coordinate) and a 𝑑-dim vector 𝑝 (generalized momentum, defined by 𝑝𝑖 = 𝜕ℒ 𝑞,𝑞ሶ,𝑡 𝜕𝑞ሶ𝑖 in physics), 𝑧 = (𝑞, 𝑝) is called the canonical coordinates. – The dynamic system is dominated by its Hamiltonian: 𝐻 𝑞, 𝑝,𝑡 = σ𝑖 𝑞ሶ 𝑖𝑝𝑖 − ℒ(𝑞, 𝑞ሶ.𝑡)ȁ𝑞ሶ=𝑞ሶ 𝑝 . If 𝑈 does not change with time, 𝐻 𝑞, 𝑝,𝑡 = 𝐻(𝑞, 𝑝), in which case 𝐻 𝑞, 𝑝 = 𝑈 + 𝐾 is the energy of the system