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Hamiltonian Mechanics

Field: Analytical Mechanics

A reformulation of classical mechanics.

Sequence of Expressions

Consider a physical system whose dynamics are described by the Lagrangian L(q,q˙,t)L(q, \dot{q}, t). The Hamiltonian formalism reformulates the dynamics by defining the conjugate momenta pip_i via the Legendre transformation:\n\npi=∂L∂q˙ip_i = \frac{\partial L}{\partial \dot{q}_i}\n\nSubsequently, the Hamiltonian function HH is defined as:\n\nH(q,p,t)=∑i=1Npiq˙i−L(q,q˙,t)H(q, p, t) = \sum_{i=1}^{N} p_i \dot{q}_i - L(q, \dot{q}, t) \n\nThe evolution is then governed by Hamilton's canonical equations.
Let (q,p)(q, p) and (Q,P)(Q, P) be two sets of canonical coordinates on a phase space M\mathcal{M}. The transformation is canonical if the Poisson bracket structure is preserved, i.e., ∂Q∂q∂P∂p−∂Q∂p∂P∂q=1\frac{\partial Q}{\partial q} \frac{\partial P}{\partial p} - \frac{\partial Q}{\partial p} \frac{\partial P}{\partial q} = 1. Equivalently, the differential form dp∧dq\text{d}p \wedge \text{d}q is invariant, and the new Hamiltonian K(Q,P,t)K(Q, P, t) is related to the old Hamiltonian H(q,p,t)H(q, p, t) by K=H+∂F∂tK = H + \frac{\partial F}{\partial t}, where FF is the generating function.
Let H(q,p,t):R2N→RH(q, p, t): \mathbb{R}^{2N} \to \mathbb{R} be the Hamiltonian function, where q=(q1,…,qN)q = (q_1, \dots, q_N) and p=(p1,…,pN)p = (p_1, \dots, p_N) are the generalized coordinates and momenta, respectively. The time evolution of the system's phase space coordinates (q,p)(\mathbf{q}, \mathbf{p}) is governed by Hamilton's canonical equations:\n\ndqidt=∂H∂pianddpidt=−∂H∂qifor i=1,…,N\frac{d q_i}{d t} = \frac{\partial H}{\partial p_i} \quad \text{and} \quad \frac{d p_i}{d t} = -\frac{\partial H}{\partial q_i} \quad \text{for } i = 1, \dots, N
Define the Poisson bracket {⋅,⋅}\left\{ \cdot , \cdot \right\} for two smooth functions f(q,p,t)f(q, p, t) and g(q,p,t)g(q, p, t) on the phase space M\mathcal{M} as:\n\n{f,g}=∑i=1N(∂f∂qi∂g∂pi−∂f∂pi∂g∂qi)\left\{ f , g \right\} = \sum_{i=1}^{N} \left( \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} \right) \n\nFor the specific case of canonical coordinates xx and yy, the bracket simplifies to {x,y}=∂x∂y−∂y∂x\left\{ x , y \right\} = \frac{\partial x}{\partial y} - \frac{\partial y}{\partial x}. The time evolution of an observable ff is given by dfdt={f,H}+∂f∂t\frac{d f}{d t} = \left\{ f , H \right\} + \frac{\partial f}{\partial t}.
Given a system with Lagrangian L(q,q˙,t)L(q, \dot{q}, t), the Hamiltonian function H(q,p,t)H(q, p, t) is defined through the Legendre transformation relating generalized coordinates qiq_i and conjugate momenta pip_i: \n\nH(q,p,t)=∑i=1Npiq˙i−L(q,q˙,t)H(q, p, t) = \sum_{i=1}^{N} p_i \dot{q}_i - L(q, \dot{q}, t) \n\nWhere the generalized velocities q˙i\dot{q}_i are implicitly determined by the momenta pip_i via the relation pi=∂L∂q˙ip_i = \frac{\partial L}{\partial \dot{q}_i}. Thus, HH is expressed solely in terms of (q,p,t)(q, p, t).
Let L(q,dqdt,t)L(q, \frac{\text{d}q}{\text{d}t}, t) be the Lagrangian of a system, and let q∈Rn\mathbf{q} \in \mathbb{R}^n be the generalized coordinates. If the action S=∫LdtS = \int L \text{d}t is invariant under a continuous transformation parameterized by ϵ\epsilon, such that δL=ddt(ϵ⋅dF/dt)\delta L = \frac{\text{d}}{\text{d}t} (\epsilon \cdot \text{d}F/\text{d}t), then there exists a conserved quantity GG (the Noether charge) defined by the generalized momentum associated with the symmetry, such that dGdt=0\frac{\text{d}G}{\text{d}t} = 0.
Given the Lagrangian L(q,q˙,t)L(q, \dot{q}, t) and the generalized momentum pi=∂L∂q˙ip_i = \frac{\partial L}{\partial \dot{q}_i}, the Hamiltonian H(q,p,t)H(q, p, t) is obtained from the Legendre transformation: H(q,p,t)=∑i=1npiq˙i−L(q,q˙,t)H(q, p, t) = \sum_{i=1}^n p_i \dot{q}_i - L(q, \dot{q}, t) where q˙i\dot{q}_i is implicitly defined by the relation pi=∂L∂q˙ip_i = \frac{\partial L}{\partial \dot{q}_i}. Furthermore, the transformation of the time derivative is dHdt=∂L∂t+∂∂t(∑piq˙i−L)\frac{\text{d}H}{\text{d}t} = \frac{\partial L}{\partial t} + \frac{\partial}{\partial t} \left( \sum p_i \dot{q}_i - L \right).
Let H(q,p,t)H(q, p, t) be the Hamiltonian of a system, and let S(q,β,t)S(q, \boldsymbol{\beta}, t) be the principal function, where β\boldsymbol{\beta} are the constants of integration (or initial momenta). The function SS must satisfy the first-order partial differential equation: H(q,∂S∂q,t)+∂S∂t=0H(q, \frac{\partial S}{\partial q}, t) + \frac{\partial S}{\partial t} = 0 This equation determines the time evolution of the system's action in the phase space.
Define the phase space M\mathcal{M} as the 2N2N-dimensional manifold parameterized by the generalized coordinates q=(q1,…,qN)\mathbf{q} = (q_1, \dots, q_N) and the conjugate momenta p=(p1,…,pN)\mathbf{p} = (p_1, \dots, p_N). A point x∈M\mathbf{x} \in \mathcal{M} represents the instantaneous state of the system, such that x=(q1,…,qN,p1,…,pN)\mathbf{x} = (q_1, \dots, q_N, p_1, \dots, p_N). The dynamics are described by the flow dxdt=J⋅∇H(x)\frac{d\mathbf{x}}{d t} = \mathbf{J} \cdot \nabla H(\mathbf{x}), where J\mathbf{J} is the symplectic matrix.