Hamiltonian Mechanics

Prerequisites: Hamiltonian Formalism, Symplectic Structure. Full path (16 pages)

• Intuitive
• Concrete
• Abstract

see also hamiltonian_formalism and hamiltons_equations

Intuitive

Hamiltonian mechanics describes a system by where it is and how much momentum it carries. Instead of tracking positions and velocities and asking what forces act, we collect all positions and all momenta into one point of a larger space, the phase space. One number attached to each point of that space, the Hamiltonian, which for most systems is the total energy, then tells the point how to move.

A picture that helps is a hiking map with contour lines of equal energy. The state of the system is a dot on the map. The Hamiltonian makes the dot slide along a contour line, so the energy stays the same, and the steepness of the terrain sets how fast it slides. For a pendulum the contours are closed loops for small swings and open wavy lines for a pendulum that goes over the top. Reading off which kind of contour you are on tells you the motion without solving anything.

The reason to use this description is that it treats positions and momenta on equal footing. That makes it easy to change to cleverer variables (canonical transformations), to see conservation laws, to describe a cloud of many possible states at once as in statistical mechanics, and to pass to quantum mechanics, where positions and momenta become operators.

Concrete

A system with $n$ degrees of freedom is described by generalized coordinates $q^a$ and conjugate momenta $p_a$, $a = 1,\dots,n$. Starting from a Lagrangian $L(q,\dot q,t)$ the momenta are $p_a = \partial L/\partial \dot q^a$, and the Hamiltonian is the Legendre transform $$H(q,p,t) = p_a \dot q^a - L(q,\dot q,t),$$ where $\dot q^a$ is expressed through $q$ and $p$ (this needs the matrix $\partial^2 L/\partial \dot q^a \partial \dot q^b$ to be invertible). The equations of motion are Hamilton's equations $$\dot q^a = \frac{\partial H}{\partial p_a}, \qquad \dot p_a = -\frac{\partial H}{\partial q^a},$$ $2n$ first-order equations in place of $n$ second-order Euler-Lagrange equations. Along a solution $dH/dt = \partial H/\partial t$, so $H$ is conserved when it has no explicit time dependence.

For $L = \frac12 m \dot{\vec x}^2 - V(\vec x)$ we get $\vec p = m\dot{\vec x}$ and $H = \frac{\vec p^2}{2m} + V(\vec x)$. Hamilton's equations give $\dot{\vec x} = \vec p/m$ and $\dot{\vec p} = -\nabla V$, which together are Newton's law.

Example: the pendulum. For a pendulum of mass $m$ and length $l$ with angle $\theta$, $L = \frac12 m l^2 \dot\theta^2 + mgl\cos\theta$, so $p_\theta = ml^2\dot\theta$ and $$H = \frac{p_\theta^2}{2ml^2} - mgl\cos\theta .$$ Since $H$ is conserved, every trajectory lies on a curve $H(\theta,p_\theta) = E$ in the $(\theta,p_\theta)$ plane. For $-mgl < E < mgl$ these curves are closed (oscillation), for $E > mgl$ they wind around in $\theta$ (rotation), and $E = mgl$ is the separatrix through the unstable point $\theta = \pi$.

Poisson brackets. With the convention $$\{f,g\} = \frac{\partial f}{\partial q^a}\frac{\partial g}{\partial p_a} - \frac{\partial f}{\partial p_a}\frac{\partial g}{\partial q^a},$$ the time evolution of any function $f(q,p,t)$ along a solution is $$\frac{df}{dt} = \{f,H\} + \frac{\partial f}{\partial t},$$ and $\{q^a,p_b\} = \delta^a_b$. A function with no explicit time dependence is conserved exactly when its bracket with $H$ vanishes. Some books, including the Abstract section below, use the opposite overall sign for the bracket; then $df/dt = \{H,f\} + \partial f/\partial t$.

Canonical transformations are changes of variables $(q,p) \to (Q,P)$ that preserve the brackets, $\{Q^a,P_b\} = \delta^a_b$, $\{Q^a,Q^b\} = \{P_a,P_b\} = 0$. They keep Hamilton's equations in the same form with a transformed Hamiltonian. Finding one in which the new Hamiltonian depends only on the momenta $P$ solves the problem, since then $P$ is constant and $Q$ grows linearly in time. This is the idea behind action-angle variables and the Hamilton-Jacobi equation.

Reading Recommendations

• The best book on Hamiltonian mechanics is The Lazy Universe by Coopersmith
• No-Nonsense Classical Mechanics by Jakob Schwichtenberg, chapter "Hamiltonian Mechanics"

Abstract

Lagrangian mechanics can be formulated geometrically using fibre bundles.

The Hamiltonian function is defined on the cotangent bundle $T^\star(C)$, which is called phase space.

In contrast, the Lagrangian function is defined on the tangent bundle $T(C)$ of the configuration space $C$.

The map from $T^\star(C) \leftrightarrow T(C)$ is called Legendre transformation.

The phase space is endowed with a symplectic structure, which defines the Poisson Bracket. The Poisson Bracket is an operation that eats two scalar fields $\Phi$, $\Psi$ on the manifold and spits out another scalar field $\theta$:

$$\theta = \{ \Phi,\Psi \}= \frac{\partial \Phi}{\partial p_a}\frac{\partial \Psi}{\partial q^a}-\frac{\partial \Phi}{\partial q^a}\frac{\partial \Psi}{\partial p_a}.$$

With this sign convention $\{q^a, p_b\} = -\delta^a_b$; many texts use the opposite sign.

If we leave the $\Psi$ slot blank, we can use the Poisson bracket to define a differential operator $\{\Phi,\ \}$. This is a vector field and when it acts on $\Psi$, we get $\{\Phi, \Psi \}$. If we use instead of $\Phi$, the Hamiltonian $H$, we get an differential operator $\{H,\ \}$ that 'points along' the trajectories in phase space $T^\star(C)$ and describes exactly the evolution that we get from Hamilton's equations.

In this sense, the dynamical evolution of a given system is completely described by the Hamiltonian (= a scalar function).

• Lectures on Mechanics by Marsden
• See page 471 in Road to Reality by R. Penrose, page 167 in Geometric Methods of Mathematical Physics by B. Schutz and http://philsci-archive.pitt.edu/2362/1/Part1ButterfForBub.pdf.
• For some more backinfo why there is a symplectic structure in classical mechanics, have a look at Classical Mechanics versus Thermodynamics (Part 2) | Azimuth

a ‘Hamiltonian’ $$H : T^* Q \to \mathbb{R}$$ or a ‘Lagrangian’ $$L : T Q \to \mathbb{R}$$

Instead, we started with Hamilton’s principal function $$S : Q \to \mathbb{R}$$ where $Q$ is not the usual configuration space describing possible positions for a particle, but the ‘extended’ configuration space, which also includes time. Only this way do Hamilton’s equations, like the Maxwell relations, become a trivial consequence of the fact that partial derivatives commute.

Classical Mechanics versus Thermodynamics (Part 2) | Azimuth

Path to this page

1. Calculus
2. Linear Algebra
3. Functional
4. Legendre Transformation
5. Manifold
6. Newton's Second Law
7. Vector Calculus
8. Differential Forms
9. Newtonian Mechanics
10. Variational Calculus
11. Configuration Space
12. Euler-Lagrange Equations
13. Lagrangian Formalism
14. Phase Space
15. Hamiltonian Formalism
16. Symplectic Structure