12.2 Hamiltonian mechanics on manifolds

Hamiltonian mechanics on manifolds provides a geometric framework for studying classical mechanics. It uses symplectic manifolds to represent phase spaces, with Hamiltonian vector fields generating dynamics. This approach emphasizes conserved quantities and symmetries in physical systems.

Key concepts include symplectic forms, Poisson brackets, and canonical transformations. Hamilton's equations describe system evolution on symplectic manifolds. Understanding this geometric perspective illuminates the deep structure underlying classical mechanics and its connection to modern physics.

Symplectic manifolds

• Symplectic manifolds provide the geometric framework for studying Hamiltonian mechanics, a formulation of classical mechanics that emphasizes the role of conserved quantities and symmetries
• A symplectic manifold is an even-dimensional smooth manifold equipped with a closed, non-degenerate 2-form called the symplectic form, which encodes the geometric structure necessary for Hamiltonian dynamics

Definition of symplectic manifold

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• A symplectic manifold is a pair $(M, \omega)$ where $M$ is a smooth manifold and $\omega$ is a symplectic form on $M$
• A symplectic form $\omega$ is a closed $(d\omega = 0)$, non-degenerate 2-form, meaning that for every non-zero tangent vector $v \in T_pM$, there exists a tangent vector $w \in T_pM$ such that $\omega(v, w) \neq 0$
• The non-degeneracy condition implies that symplectic manifolds are always even-dimensional

Examples of symplectic manifolds

• The cotangent bundle $T^*M$ of any smooth manifold $M$ is a symplectic manifold with the canonical symplectic form $\omega = \sum_{i=1}^n dq_i \wedge dp_i$, where $(q_i, p_i)$ are local coordinates on $T^*M$
• The product of two symplectic manifolds $(M_1, \omega_1)$ and $(M_2, \omega_2)$ is a symplectic manifold with the symplectic form $\omega = \pi_1^*\omega_1 + \pi_2^*\omega_2$, where $\pi_1$ and $\pi_2$ are the projection maps
• Kähler manifolds, which are complex manifolds with a compatible Riemannian metric and symplectic form, are examples of symplectic manifolds (complex projective spaces)

Darboux's theorem

• Darboux's theorem states that locally, all symplectic manifolds of the same dimension are indistinguishable
• More precisely, for any point $p$ in a symplectic manifold $(M, \omega)$, there exists a neighborhood $U$ of $p$ and local coordinates $(q_1, \dots, q_n, p_1, \dots, p_n)$ on $U$ such that $\omega|_U = \sum_{i=1}^n dq_i \wedge dp_i$
• This theorem implies that symplectic manifolds have no local invariants other than the dimension, and the study of symplectic manifolds focuses on their global properties

Hamiltonian vector fields

• Hamiltonian vector fields are the fundamental objects that generate the dynamics on a symplectic manifold
• They are defined in terms of a smooth function called the Hamiltonian, which represents the total energy of the system

Definition of Hamiltonian vector field

• Given a smooth function $H: M \to \mathbb{R}$ on a symplectic manifold $(M, \omega)$, the Hamiltonian vector field $X_H$ associated with $H$ is the unique vector field satisfying $dH = \omega(X_H, \cdot)$
• In local coordinates $(q_i, p_i)$, the Hamiltonian vector field is given by $X_H = \sum_{i=1}^n \left(\frac{\partial H}{\partial p_i} \frac{\partial}{\partial q_i} - \frac{\partial H}{\partial q_i} \frac{\partial}{\partial p_i}\right)$

Properties of Hamiltonian vector fields

• Hamiltonian vector fields preserve the symplectic form: $\mathcal{L}_{X_H}\omega = 0$, where $\mathcal{L}_{X_H}$ denotes the Lie derivative along $X_H$
• The flow generated by a Hamiltonian vector field consists of symplectomorphisms, which are diffeomorphisms that preserve the symplectic form
• The Poisson bracket of two functions $f$ and $g$ on a symplectic manifold is related to the Lie bracket of their Hamiltonian vector fields: $\{f, g\} = \omega(X_f, X_g)$

Poisson brackets

• The Poisson bracket is a bilinear operation on smooth functions that encodes the geometry of the symplectic manifold and the algebra of observables in Hamiltonian mechanics
• Given two smooth functions $f, g: M \to \mathbb{R}$, their Poisson bracket is defined as $\{f, g\} = \omega(X_f, X_g)$
• In local coordinates $(q_i, p_i)$, the Poisson bracket is given by $\{f, g\} = \sum_{i=1}^n \left(\frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i}\right)$
• The Poisson bracket satisfies the Jacobi identity: $\{f, \{g, h\}\} + \{g, \{h, f\}\} + \{h, \{f, g\}\} = 0$, making the space of smooth functions a Lie algebra under the Poisson bracket operation

Hamiltonian dynamics

• Hamiltonian dynamics describes the time evolution of a physical system in terms of the symplectic geometry of its phase space, which is typically the cotangent bundle of the configuration manifold
• The dynamics are governed by Hamilton's equations, which are first-order differential equations involving the Hamiltonian function

Hamilton's equations

• Given a Hamiltonian function $H: M \to \mathbb{R}$ on a symplectic manifold $(M, \omega)$, Hamilton's equations are a system of first-order differential equations that describe the time evolution of the system: $\frac{dq_i}{dt} = \frac{\partial H}{\partial p_i}, \quad \frac{dp_i}{dt} = -\frac{\partial H}{\partial q_i}$

• The solutions to Hamilton's equations are integral curves of the Hamiltonian vector field $X_H$, which generate the Hamiltonian flow on the symplectic manifold

Conservation of energy

• A key feature of Hamiltonian systems is the conservation of energy along the Hamiltonian flow
• If the Hamiltonian $H$ does not explicitly depend on time, then $\frac{dH}{dt} = 0$ along the solutions of Hamilton's equations
• This means that the Hamiltonian function, which typically represents the total energy of the system, remains constant along the trajectories of the system

Liouville's theorem

• Liouville's theorem states that the Hamiltonian flow preserves the symplectic volume form $\omega^n = \omega \wedge \cdots \wedge \omega$ ($n$ times) on the symplectic manifold
• As a consequence, the volume of any region in the phase space remains constant under the Hamiltonian flow
• Liouville's theorem is a manifestation of the incompressibility of the Hamiltonian flow and has important implications for statistical mechanics and the study of ensembles of Hamiltonian systems

Canonical transformations

• Canonical transformations are coordinate transformations on a symplectic manifold that preserve the symplectic structure
• They play a crucial role in simplifying the analysis of Hamiltonian systems and in the study of symmetries and conserved quantities

Definition of canonical transformation

• A canonical transformation is a diffeomorphism $\phi: M \to M$ on a symplectic manifold $(M, \omega)$ that preserves the symplectic form: $\phi^*\omega = \omega$
• Equivalently, a canonical transformation is a coordinate transformation $(q_i, p_i) \mapsto (Q_i, P_i)$ that preserves the form of Hamilton's equations: $\frac{dQ_i}{dt} = \frac{\partial K}{\partial P_i}, \quad \frac{dP_i}{dt} = -\frac{\partial K}{\partial Q_i}$

where $K(Q_i, P_i, t)$ is the transformed Hamiltonian

Generating functions

• Canonical transformations can be described using generating functions, which are scalar functions that implicitly define the transformation
• There are four common types of generating functions, each depending on a different combination of old and new coordinates:

1. $F_1(q_i, Q_i, t)$: $p_i = \frac{\partial F_1}{\partial q_i}, \quad P_i = -\frac{\partial F_1}{\partial Q_i}$
2. $F_2(q_i, P_i, t)$: $p_i = \frac{\partial F_2}{\partial q_i}, \quad Q_i = \frac{\partial F_2}{\partial P_i}$
3. $F_3(p_i, Q_i, t)$: $q_i = -\frac{\partial F_3}{\partial p_i}, \quad P_i = -\frac{\partial F_3}{\partial Q_i}$
4. $F_4(p_i, P_i, t)$: $q_i = -\frac{\partial F_4}{\partial p_i}, \quad Q_i = \frac{\partial F_4}{\partial P_i}$

• The transformed Hamiltonian $K$ is related to the original Hamiltonian $H$ and the generating function $F$ by the equation: $K(Q_i, P_i, t) = H(q_i, p_i, t) + \frac{\partial F}{\partial t}$

Examples of canonical transformations

• Point transformations, which are coordinate transformations that depend only on the coordinates $(q_i, Q_i)$, are canonical transformations with generating functions of type $F_1$
• The Legendre transformation, which relates the Lagrangian and Hamiltonian formulations of mechanics, is a canonical transformation with a generating function of type $F_2$
• The action-angle transformation, which is used in the study of integrable systems, is a canonical transformation that separates the dynamics into periodic motions and conserved quantities

Symplectic group

• The symplectic group is the group of linear canonical transformations on a symplectic vector space
• It plays a fundamental role in the study of the symmetries and conservation laws of Hamiltonian systems

Definition of symplectic group

• The symplectic group $Sp(2n, \mathbb{R})$ is the group of linear transformations $A$ on a symplectic vector space $(V, \omega)$ of dimension $2n$ that preserve the symplectic form: $\omega(Av, Aw) = \omega(v, w)$ for all $v, w \in V$
• In matrix form, a symplectic matrix $A$ satisfies the condition $A^T J A = J$, where $J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}$ is the standard symplectic matrix and $I_n$ is the $n \times n$ identity matrix

Lie algebra of symplectic group

• The Lie algebra $\mathfrak{sp}(2n, \mathbb{R})$ of the symplectic group consists of the matrices $X$ satisfying $X^T J + J X = 0$
• The elements of the Lie algebra generate infinitesimal symplectic transformations and are closely related to the conserved quantities of Hamiltonian systems
• The Lie algebra $\mathfrak{sp}(2n, \mathbb{R})$ is isomorphic to the space of quadratic polynomials in $q_i$ and $p_i$ under the Poisson bracket operation

Symplectic matrices

• Symplectic matrices have several important properties:
  • The inverse of a symplectic matrix is also symplectic: $(A^T J A = J) \Rightarrow (A^{-1} = -J A^T J)$
  • The product of two symplectic matrices is symplectic: $(A^T J A = J, B^T J B = J) \Rightarrow ((AB)^T J (AB) = J)$
  • The determinant of a symplectic matrix is equal to 1: $\det(A) = 1$
• Examples of symplectic matrices include rotation matrices in the plane, which have the form $\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$, and the Pauli matrices $\sigma_x$, $\sigma_y$, and $\sigma_z$ in quantum mechanics

Hamiltonian systems on cotangent bundles

• Many physical systems, such as particles moving in a potential or rigid body dynamics, are naturally described as Hamiltonian systems on the cotangent bundle of a configuration manifold
• The cotangent bundle has a canonical symplectic structure that makes it a natural setting for Hamiltonian mechanics

Cotangent bundles as symplectic manifolds

• Given a smooth manifold $Q$, its cotangent bundle $T^*Q$ is the space of covectors (linear functionals) on the tangent spaces of $Q$
• The cotangent bundle $T^*Q$ has a canonical symplectic form $\omega = dq_i \wedge dp_i$, where $(q_i, p_i)$ are local coordinates on $T^*Q$ induced by coordinates $q_i$ on $Q$
• The canonical symplectic form makes $T^*Q$ a symplectic manifold, and the projection $\pi: T^*Q \to Q$ is called the cotangent bundle projection

Legendre transform

• The Legendre transform is a canonical transformation that relates the Lagrangian and Hamiltonian formulations of mechanics

• Given a Lagrangian function $L(q_i, \dot{q}_i, t)$, the Legendre transform defines the conjugate momenta $p_i$ and the Hamiltonian function $H(q_i, p_i, t)$ as: $p_i = \frac{\partial L}{\partial \dot{q}_i}, \quad H(q_i, p_i, t) = p_i \dot{q}_i - L(q_i, \dot{q}_i, t)$

• The Legendre transform is a canonical transformation with a generating function of type $F_2(q_i, P_i, t) = P_i \dot{q}_i - L(q_i, \dot{q}_i, t)$, where $P_i$ are the new momenta

Hamiltonian formulation of Lagrangian mechanics

• The Hamiltonian formulation of mechanics is obtained by applying the Legendre transform to the Lagrangian formulation
• The Euler-Lagrange equations of motion in Lagrangian mechanics, $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0,$

are transformed into Hamilton's equations in the Hamiltonian formulation:

$\frac{dq_i}{dt} = \frac{\partial H}{\partial p_i}, \quad \frac{dp_i}{dt} = -\frac{\partial H}{\partial q_i}$

• The Hamiltonian formulation provides a more symmetric treatment of coordinates and momenta and is better suited for studying symmetries, conservation laws, and canonical transformations

Moment maps

• Moment maps are a generalization of conserved quantities in the presence of symmetries, and they play a crucial role in the study of Hamiltonian systems with symmetries
• They provide a link between the symmetries of a Hamiltonian system and the geometry of the symplectic manifold

Definition of moment map

• Let $$(M, \