Lagrangian and Hamiltonian Systems

Introduction

A Lagrangian system is a mechanical system described by a function on the tangent bundle of a configuration space, and its equation is the Euler–Lagrange equation of the action integral; a Hamiltonian system is the same system described by a function on the cotangent bundle, and its equation is the first-order system defined by a symplectic form. The passage between the two descriptions is the Legendre transform, which pairs velocities with momenta, and the two pictures are equivalent whenever the transform is invertible. This article develops that equivalence and the structure of each picture: the Lagrangian mechanics, with its variational principle, the mechanical origin of its equation in D'Alembert's principle, the non-holonomic constraints and their multipliers, its symmetries and its Noether conservation laws; the symplectic and Poisson formulation of Hamiltonian mechanics, with its bracket formulation and Poisson's theorem, its canonical transformations and generating functions, and its infinitesimal canonical transformations generated by the flow; the modified Hamilton principle, in which the coordinates and the momenta are varied independently; the conservation laws of the Hamiltonian flow, among them the preservation of phase volume, Liouville's equation for the density and the recurrence theorem; the finite-dimensional theory of complete integrability, with the Liouville–Arnold theorem, invariant tori and action-angle variables; and the classical examples — the oscillator, the pendulum and its phase portrait, the Kepler problem, the rigid body and the geodesic flow.

The article is the finite-dimensional counterpart of the theory of integrable systems of the two preceding articles. There the integrability is that of an infinite-dimensional system, detected by a Lax pair and solved by an inverse scattering transform; here it is the integrability of a finite-dimensional Hamiltonian system, detected by a supply of independent integrals in involution and solved by quadrature, and the two theories meet in the travelling-wave reductions of the integrable equations, which are finite-dimensional Hamiltonian systems. The variational statement of the equation of the system — the Euler–Lagrange equation, the Legendre transform, the Hamilton–Jacobi equation and Noether's theorem — has been established in the article of this Part on the calculus of variations, and is used here rather than re-derived; the symplectic form and the Poisson bracket are the structures of Part I, developed in Symplectic Forms and Poisson Brackets, and the geometry of symplectic manifolds is the subject of Part II, written in parallel.

Lagrangian Systems

The Variational Principle

Definition. Let $Q$ be a smooth manifold, the configuration space, with tangent bundle $TQ$ and coordinates $(q,\dot q)$; let $L : TQ\times\mathbb{R}\to\mathbb{R}$ be a smooth Lagrangian, $L=L(q,\dot q,t)$. The action of a curve $q : [t_0,t_1]\to Q$ with fixed endpoints is

$$ S[q] = \int_{t_0}^{t_1}L(q(t),\dot q(t),t)\,dt , $$

and the Euler–Lagrange equations of $L$ are

$$ \frac{d}{dt}\frac{\partial L}{\partial\dot q^i} - \frac{\partial L}{\partial q^i} = 0, \qquad i = 1,\dots,n . $$

Theorem (Hamilton's principle). A curve $q$ is a critical point of the action among curves with fixed endpoints if and only if it satisfies the Euler–Lagrange equations.

Proof. This is the first variation of the functional of the calculus of variations, applied to the Lagrangian $L(q,\dot q,t)$ with the roles of the independent variable $x$ and the field $u$ taken by $t$ and $q$; the derivation, the du Bois-Reymond argument and the natural boundary conditions are those of that article.

Definition. The system is autonomous if $L$ is independent of $t$. For an autonomous Lagrangian the energy is

$$ E = \sum_{i=1}^n\dot q^i\frac{\partial L}{\partial\dot q^i} - L , $$

and it is constant along every solution, by the Beltrami identity.

Virtual Work, D'Alembert's Principle and Non-Holonomic Constraints

The variational principle takes the Lagrangian as given and produces the equation; the mechanical origin of the equation is a different statement, and it is the one that fixes the domain of validity. D'Alembert's principle is Newton's law written in terms of the work done on the displacements that the constraints permit, and the Lagrange equation is its transcription into generalised coordinates. The distinction matters because the principle continues to apply where the action is not stationary, which is the non-holonomic case.

Definition. Let a system consist of $\mathcal N$ constituents at positions $\mathbf r_1,\dots,\mathbf r_{\mathcal N}$ in three-dimensional space, subject to $\kappa$ independent constraints. A virtual displacement $(\delta\mathbf r_1,\dots,\delta\mathbf r_{\mathcal N})$ is an infinitesimal displacement consistent with the constraints at fixed time; the constraints are ideal when the virtual work of the constraint forces vanishes, $\sum_a\mathbf R_a\cdot\delta\mathbf r_a=0$ for every virtual displacement. The number of degrees of freedom is $3\mathcal N-\kappa$, the number of generalised coordinates $q^1,\dots,q^n$ is equal to it, and the positions are expressed as $\mathbf r_a=\mathbf r_a(q,t)$.

Principle (D'Alembert). The impressed forces $\mathbf F_a$ and the accelerations satisfy

$$ \sum_a\bigl(\mathbf F_a-m_a\ddot{\mathbf r}_a\bigr)\cdot\delta\mathbf r_a = 0 $$

for every virtual displacement. The constraint forces do not appear, because they are ideal; the principle is Newton's law with the constraint forces eliminated rather than solved for.

Theorem (D'Alembert's principle gives the Lagrange equation). Let the virtual work of the impressed forces be $Q_j\,\delta q^j$ in generalised coordinates, so that $Q_j$ is the generalised force. Then D'Alembert's principle is equivalent to

$$ \frac{d}{dt}\frac{\partial T}{\partial\dot q^j}-\frac{\partial T}{\partial q^j} = Q_j , \qquad j=1,\dots,n , $$

with $T$ the kinetic energy. If the forces are conservative with potential $V$, so that $Q_j=-\partial V/\partial q^j$, the equation is the Euler–Lagrange equation of $L=T-V$.

Proof. The displacement of the $a$-th constituent is $\delta\mathbf r_a=(\partial\mathbf r_a/\partial q^j)\delta q^j$, so the virtual work of the impressed forces is $Q_j\delta q^j$ with $Q_j=\sum_a\mathbf F_a\cdot\partial\mathbf r_a/\partial q^j$. For the inertial term, the product rule gives

$$ \sum_am_a\ddot{\mathbf r}_a\cdot\frac{\partial\mathbf r_a}{\partial q^j} = \frac{d}{dt}\Bigl(\sum_am_a\dot{\mathbf r}_a\cdot\frac{\partial\mathbf r_a}{\partial q^j}\Bigr)-\sum_am_a\dot{\mathbf r}_a\cdot\frac{d}{dt}\frac{\partial\mathbf r_a}{\partial q^j} , $$

and with $T=\tfrac12\sum_am_a\lvert\dot{\mathbf r}_a\rvert^2$ the identities $\partial\mathbf r_a/\partial q^j=\partial\dot{\mathbf r}_a/\partial\dot q^j$ and $\frac{d}{dt}\partial\mathbf r_a/\partial q^j=\partial\dot{\mathbf r}_a/\partial q^j$ make the two terms $\frac{d}{dt}\partial T/\partial\dot q^j$ and $\partial T/\partial q^j$. ∎

Definition. A constraint is holonomic if it is the vanishing of a relation $\phi(q,t)=0$ among the coordinates; it is scleronomic when $\phi$ carries no explicit time and rheonomic when it does. A holonomic constraint is used by elimination: it removes one coordinate, and the remaining coordinates are free. A constraint that is linear in the velocities, $A^a_j(q)\dot q^j=0$, and that admits no integrating factor is non-holonomic; it cannot be eliminated, and it must be imposed by multipliers.

Theorem (the Lagrange–d'Alembert equations). Let the constraints be $A^a_j(q)\dot q^j=0$, $a=1,\dots,\kappa$, with the rows of $A$ independent, so that the admissible virtual displacements are the vectors with $A^a_j\delta q^j=0$. Then D'Alembert's principle is equivalent to

$$ \frac{d}{dt}\frac{\partial L}{\partial\dot q^j}-\frac{\partial L}{\partial q^j} = \lambda_aA^a_j , \qquad A^a_j\dot q^j=0 , $$

the $\lambda_a$ being functions of $t$ determined together with the solution.

Proof. At a configuration the admissible virtual displacements form the subspace annihilated by the rows of $A$; D'Alembert's principle requires the generalised force to annihilate that same subspace, so it is a combination $\lambda_aA^a_j$ of the rows, by the independence of the rows. The constraint is imposed on the velocity rather than on the virtual displacement, and the difference is not merely formal: imposing the same relation on the variation of the action instead gives the vakonomic equations, which agree with these for holonomic constraints and differ from them otherwise, and it is D'Alembert's principle, not stationary action, that reproduces Newton's law. The pair of equations is the differential-algebraic system of the article of this Part on differential-algebraic equations. ∎

Example (the skate, or knife edge). A body moving in the plane with position $(x,y)$ and heading $\theta$, whose velocity is along its heading, satisfies

$$ \dot x\sin\theta-\dot y\cos\theta = 0 . $$

The annihilator of the distribution is the one-form $\omega=\sin\theta\,dx-\cos\theta\,dy$, and

$$ \omega\wedge d\omega = -\,d\theta\wedge dx\wedge dy \neq 0 , $$

so by the Frobenius theorem the distribution is not integrable and the constraint admits no relation $\phi(x,y,\theta)=0$: the body cannot move sideways by an admissible motion, yet a closed loop of admissible motions shifts its position, which is parallel parking. With $L=\tfrac12(\dot x^2+\dot y^2)+\tfrac12I\dot\theta^2$ the Lagrange–d'Alembert equations are

$$ \ddot x = \lambda\sin\theta , \qquad \ddot y = -\lambda\cos\theta , \qquad I\ddot\theta = 0 , $$

the acceleration of the contact point lying along the heading and the heading turning uniformly; the multiplier $\lambda$ is the non-holonomic force that keeps the velocity along the heading, and it is fixed by the constraint together with the solution rather than given in advance.

Theorem (the energy of a non-holonomic system). For an autonomous Lagrangian and a constraint that is homogeneous in the velocities, the energy $E=\dot q^j\partial L/\partial\dot q^j-L$ is conserved along the solutions of the Lagrange–d'Alembert equations, even though the system has no stationary-action principle.

Proof. Multiply the equation by $\dot q^j$ and sum over $j$: the multiplier term contributes $\lambda_aA^a_j\dot q^j=0$, by the constraint. Hence $\dot q^j\bigl(\frac{d}{dt}L_{\dot q^j}-L_{q^j}\bigr)=0$, and the left-hand side is $\frac{d}{dt}E+L_t$; with $L_t=0$ the energy is constant. The computation is the one that gives the Beltrami identity, with the multiplier term killed by the constraint rather than absent. ∎

The multiplier structure — an unknown function determined by the constraint rather than eliminated — is the same one that the constrained problem of the calculus of variations exhibits, and it is the finite-dimensional form of the costate of optimal control, where the constraint is a differential equation for the state rather than a velocity relation; the article of this Part on optimal control treats that case, and the same multiplier rule produces the costate there.

Mechanical Similarity

Proposition (mechanical similarity). Take $Q=\mathbb{R}^n$ with the Euclidean kinetic energy, $L=\tfrac12m\dot q^i\dot q_i-V(q)$, and suppose the potential is time-independent and homogeneous of degree $k$,

$$ V(\alpha q)=\alpha^kV(q) \qquad\text{for every } \alpha>0 . $$

Rescale a solution by $q(t)\mapsto\alpha q(\beta t)$. The kinetic energy acquires the factor $(\alpha/\beta)^2$ and the potential the factor $\alpha^k$, so the rescaled Lagrangian is a constant multiple of the original, $L'=\alpha^kL$, exactly when

$$ \beta=\alpha^{1-k/2} . $$

The Euler–Lagrange operator is linear in $L$, so multiplying $L$ by a nonzero constant multiplies its equation by the same constant and does not change its solutions. The rescaled curve is therefore again a solution, and the trajectories are geometrically similar: lengths scale as $\alpha$, times as $\alpha^{1-k/2}$, and the combination $t\,l^{k/2-1}$ is invariant along the family.

Two values of $k$ recover the classical landmarks.

$k=-1$ (the Kepler and Coulomb potentials, $V\propto1/r$): $\beta=\alpha^{3/2}$, so the period scales as the $3/2$ power of the size. For the family of similar orbits this is Kepler's third law, $T^2\propto a^3$.
$k=2$ (the isotropic harmonic oscillator, $V\propto r^2$): $\beta=1$, so the time scale is independent of the amplitude — the isochronism that makes the oscillator's period independent of its energy.

The homogeneity of the potential is what lets the kinetic and potential terms scale together; a potential that is a sum of two homogeneous parts of different degrees admits no such scaling.

Cyclic Coordinates and Conserved Momenta

Definition. A coordinate $q^k$ of a Lagrangian system is cyclic, or ignorable, if the Lagrangian does not depend on it, $\partial L/\partial q^k=0$. The conjugate momentum $p_k=\partial L/\partial\dot q^k$ is then the momentum conjugate to $q^k$.

Theorem (conservation of a cyclic momentum). If $q^k$ is cyclic, then $p_k$ is constant along every solution of the Euler–Lagrange equations.

Proof. The Euler–Lagrange equation with index $k$ reads $\frac{d}{dt}p_k=\frac{\partial L}{\partial q^k}$, and the right-hand side vanishes by cyclicity, so $p_k$ is a constant of the motion. The conserved quantity is the Noether charge of the one-parameter family of translations of $q^k$, which leaves the Lagrangian invariant exactly when $q^k$ is cyclic. ∎

Example (angular momentum). A particle of mass $m$ in a plane, in a central potential $V(r)$, has the Lagrangian in polar coordinates $(t,r,\theta)$ $$ L = \frac12m\bigl(\dot r^2+r^2\dot\theta^2\bigr)-V(r) . $$ The coordinate $\theta$ is cyclic, so $p_\theta=mr^2\dot\theta$ is conserved along every solution; this is the angular momentum about the origin, and its conservation is the statement that a central force exerts no torque. Each cyclic coordinate reduces the order of the system by one.

The Legendre Transform and the Hamiltonian

Definition. The Legendre transform or fibre derivative of $L$ is the map $\mathbb{F}L : TQ\to T^*Q$ given in coordinates by

$$ (q,\dot q)\mapsto (q,p), \qquad p_i = \frac{\partial L}{\partial\dot q^i}(q,\dot q) . $$

The Lagrangian is regular if $\mathbb{F}L$ is a local diffeomorphism, equivalently if the Hessian matrix $\bigl(\partial^2L/\partial\dot q^i\partial\dot q^j\bigr)$ is everywhere nonsingular; it is hyperregular if $\mathbb{F}L$ is a global diffeomorphism.

Theorem (the Hamiltonian). If $L$ is hyperregular, then the function

$$ H(q,p) = \sum_{i=1}^n p_i\dot q^i - L(q,\dot q), \qquad \dot q = (\mathbb{F}L)^{-1}(q,p) , $$

is well defined, and a curve $q$ satisfies the Euler–Lagrange equations of $L$ if and only if the curve $(q,p)=(\mathbb{F}L)(q,\dot q)$ satisfies Hamilton's equations

$$ \dot q^i = \frac{\partial H}{\partial p_i}, \qquad \dot p_i = -\frac{\partial H}{\partial q^i} . $$

Proof. The Legendre transform of the calculus of variations, computed for the case in which $L$ is convex in $\dot q$; the envelope theorem gives $H_{p_i}=\dot q^i$ and $H_{q^i} = -L_{q^i}$, so that $\dot p_i = \frac{d}{dt}L_{\dot q^i}=L_{q^i}=-H_{q^i}$ along a solution. The invertibility of $\mathbb{F}L$ makes the change of variables from $(q,\dot q)$ to $(q,p)$ a diffeomorphism.

Remark (the modified Hamilton principle). Hamilton's equations follow from a variational principle in which the coordinates and the momenta are varied independently. For the modified Lagrangian

$$ \tilde L(q,p,\dot q,\dot p) = \sum_i p_i\dot q^i - H(q,p,t) , $$

the action $\int\tilde L\,dt$ has as its Euler–Lagrange equations the first-order system

$$ \frac{d}{dt}\frac{\partial\tilde L}{\partial\dot q^i}-\frac{\partial\tilde L}{\partial q^i}=0 \quad\Longrightarrow\quad \dot p_i=-H_{q^i} , \qquad \frac{d}{dt}\frac{\partial\tilde L}{\partial\dot p_i}-\frac{\partial\tilde L}{\partial p_i}=0 \quad\Longrightarrow\quad \dot q^i=H_{p_i} , $$

because $\partial\tilde L/\partial\dot q^i=p_i$, $\partial\tilde L/\partial q^i=-H_{q^i}$, $\partial\tilde L/\partial\dot p_i=0$ and $\partial\tilde L/\partial p_i=\dot q^i-H_{p_i}$. The principle $\delta\int(\sum_ip_i\dot q^i-H)\,dt=0$, with independent variations of $q$ and $p$, is therefore an equivalent statement of the Hamiltonian equations, and it is the form in which the Hamiltonian formalism is the stationarity of an action. The two equations differ in kind: the variation in $p$ is algebraic in $\dot q$ and reproduces the Legendre transform, $\dot q^i=H_{p_i}$, while the variation in $q$ carries the dynamics. The modified principle is not the Lagrangian principle of the first part of this article, because $\tilde L$ is a function on the enlarged space of $q,p,\dot q$ and is not a Lagrangian on the tangent bundle of $Q$; its independent variation of $p$ is the feature that the ordinary principle does not have.

Example (the mechanical Lagrangian). For a Riemannian metric $g$ on $Q$ and a potential $V : Q\to\mathbb{R}$, the Lagrangian

$$ L(q,\dot q) = \frac12g_{ij}(q)\dot q^i\dot q^j - V(q) $$

is hyperregular, with $p_i = g_{ij}\dot q^j$ and

$$ H(q,p) = \frac12g^{ij}(q)p_ip_j + V(q) , $$

the sum of the kinetic and potential energies. The Euler–Lagrange equation of the pure kinetic Lagrangian, $V=0$, is the geodesic equation of $g$, so a free particle on a Riemannian manifold moves along a geodesic; the metric, the Christoffel symbols and the geodesic equation are those of Part II. The corresponding Hamiltonian flow on $T^*Q$ is the geodesic flow, whose dynamical properties are treated in the article of this Part on the geodesic flow, written in parallel.

Example (the oscillator and the pendulum). For $Q=\mathbb{R}$ and $L=\frac12\dot q^2-\frac12\omega^2q^2$ the Hamiltonian is $H=\frac12(p^2+\omega^2q^2)$ and the flow is the linear oscillator, with the level sets $H=\text{const}$ being ellipses. For the pendulum, $L = \frac12\dot q^2+\cos q$, the Hamiltonian is $H=\frac12p^2-\cos q$; the level sets with $H<1$ are closed orbits (librations), the separatrix $H=1$ joins the unstable equilibrium at $q=\pi$, and the level sets with $H>1$ are open (rotations). The example is the first place where the topological classification of the level sets of $H$ governs the qualitative dynamics.

Remark (the phase portrait in one degree of freedom). In one degree of freedom the level curves of $H$ are the whole phase portrait, and the equilibria are the critical points of $H$. At a critical point where the Hessian of $H$ is definite — a centre, or elliptic point — the nearby level curves are closed and the motion is periodic (libration); where the Hessian is indefinite — a saddle, or hyperbolic point — the level curve through the point is the separatrix, which joins the point to itself and divides the phase plane into regions of different topology. The type is read off the linearisation without the global picture: the Jacobian of the flow at a critical point of $H$ is $JH''$, with $J$ the symplectic matrix and $H''$ the Hessian, so the eigenvalues are $\pm i\sqrt{\lambda_1\lambda_2}$ when the Hessian eigenvalues $\lambda_1,\lambda_2$ have the same sign (the centre) and $\pm\sqrt{-\lambda_1\lambda_2}$ when they have opposite signs (the saddle). The pendulum is the model: $H=\frac12p^2-\cos q$ is definite at $q=0$, the stable hanging equilibrium and a centre, and indefinite at $q=\pi$, the unstable inverted equilibrium and a saddle, while the separatrix $H=1$ is the curve $p=\pm2\cos(q/2)$ joining the two saddles and separates the librations $H<1$ from the rotations $H>1$. The saddle is no exception to Liouville's theorem, since the separatrix has measure zero.

The Canonical Momentum One-Form

The Legendre transform pairs velocities with momenta, and with that pairing the equation of the system can be written without coordinates, as the vanishing of a Lie derivative.

Definition. The canonical momentum one-form of $L$ is the $1$-form on the tangent bundle $$ \theta_L = p_i\,dq^i = \frac{\partial L}{\partial\dot q^i}\,dq^i , $$ the pullback of the tautological form $p_i\,dq^i$ of the cotangent bundle by the Legendre transform. A vector field $X$ on $TQ$ is a second-order field if $X(q^i)=\dot q^i$ for each $i$, that is, if its integral curves are the velocity lifts of their projections to $Q$.

Theorem (coordinate-free Euler–Lagrange equation). Let $L$ be a regular Lagrangian, so that the Hessian $\bigl(\partial^2L/\partial\dot q^i\partial\dot q^j\bigr)$ is everywhere invertible. Then there is a unique second-order vector field $X$ on $TQ$ with $$ \mathcal{L}_X\theta_L = dL , $$ and the integral curves of $X$ are exactly the velocity lifts of the solutions of the Euler–Lagrange equations. The field $X$ is the field of the system, and the identity is the Euler–Lagrange equation in coordinate-free form.

Proof. Write $X$ with $X(q^i)=\dot q^i$ and unknown accelerations $X(\dot q^j)$, and put $\alpha_i=X(p_i)$. Cartan's formula gives $$ \mathcal{L}_X\theta_L = d(\iota_X\theta_L)+\iota_X d\theta_L = d(p_i\dot q^i)+\alpha_i\,dq^i-\dot q^i\,dp_i = p_i\,d\dot q^i+\alpha_i\,dq^i , $$ the last step using $d(p_i\dot q^i)=\dot q^i\,dp_i+p_i\,d\dot q^i$. Equating coefficients with those of $dL = L_{q^i}dq^i+L_{\dot q^i}d\dot q^i$ gives $p_i=L_{\dot q^i}$, which is the definition of $\theta_L$, and $\alpha_i=L_{q^i}$, that is $$ \frac{d}{dt}\Bigl(\frac{\partial L}{\partial\dot q^i}\Bigr)=\frac{\partial L}{\partial q^i} $$ along an integral curve of $X$, the Euler–Lagrange equation. Conversely, $\alpha_i=L_{q^i}$ is the linear system $$ \sum_j L_{\dot q^i\dot q^j}\,X(\dot q^j) = L_{q^i}-\dot q^jL_{\dot q^i q^j} $$ for the accelerations, with the Hessian as coefficient matrix; regularity makes the matrix invertible, so the accelerations, and with them $X$, are unique. ∎

Remark (the Cartan form and the field case). For a time-dependent Lagrangian the one-form is replaced on $TQ\times\mathbb{R}$ by the Poincaré–Cartan form $L\,dt+p_i(dq^i-\dot q^i\,dt)$, whose Lie derivative along the field of the system carries the same content together with the conservation of the energy. In a field theory the momentum is an $(m-1)$-form rather than a one-form, the Lie derivative is replaced by a divergence, and the local coordinate form is the multi-index equation of The Calculus of Variations. The Hamiltonian side of the field case is the subject of the article of this Part Multisymplectic and Covariant Hamiltonian Field Theory: there the single momentum is replaced by a polymomentum indexed by the directions of the independent variables, the momentum being correspondingly a family of $(m-1)$-forms, the symplectic form by a closed form of higher degree, and the two-form's invertibility by a degree count that leaves no covariant Poisson bracket. The case $m=1$ of that article is the present theory, so the remark is a forward reference and not an analogy.

Higher-Derivative Lagrangians and the Ostrogradsky Instability

Nothing above required $L$ to depend on the coordinates and their first derivatives only, and the calculus of variations extends to an integrand $L(q,\dot q,\ddot q,\dots,q^{(N)})$. The first variation now gives the higher-order Euler–Lagrange equation

$$ \sum_{j=0}^{N}(-1)^j\frac{d^j}{dt^j}\frac{\partial L}{\partial q^{(j)}}=0 , $$

each integration by parts transferring one derivative off the variation and onto $\partial L/\partial q^{(j)}$ and contributing the sign $(-1)^j$. The canonical formalism extends with it, and for $N\ge2$ it does so at a price.

Ostrogradsky's theorem. For $N=2$ set $Q^1=q$, $Q^2=\dot q$ and the two momenta

$$ p_1=\frac{\partial L}{\partial\dot q}-\frac{d}{dt}\frac{\partial L}{\partial\ddot q}, \qquad p_2=\frac{\partial L}{\partial\ddot q} . $$

When the Hessian $\partial^2L/\partial\ddot q^i\partial\ddot q^j$ is nonsingular, $\ddot q$ is solved from $p_2$, and the Hamiltonian is $H=p_1\dot q+p_2\ddot q-L$. This $H$ is linear in $p_1$, with coefficient $\dot q$: neither $L$ nor the eliminated $\ddot q$ depends on $p_1$, so $\partial H/\partial p_1=\dot q$. On every configuration with $\dot q\neq0$ the Hamiltonian is therefore unbounded below, $H\to-\infty$ along $p_1\to-\infty$ at fixed $\dot q>0$, and a non-degenerate higher-derivative Lagrangian has no stable ground state — the Ostrogradsky instability.

For example $L=\tfrac12\ddot q^2$ has $p_2=\ddot q$, $H=p_1\dot q+\tfrac12p_2^2$ and Euler–Lagrange equation $q^{(4)}=0$; the Pais–Uhlenbeck oscillator

$$ L=\tfrac12\ddot q^2-\tfrac12(\omega_1^2+\omega_2^2)\dot q^2+\tfrac12\omega_1^2\omega_2^2q^2 $$

has the same linear term, with the equation $q^{(4)}+(\omega_1^2+\omega_2^2)\ddot q+\omega_1^2\omega_2^2q=0$.

The instability is a statement about the order of the time derivatives, not about the number of derivatives as such: an integrand whose additional derivatives are spatial only leaves the time-derivative structure of the ordinary kinetic term intact and does not produce the linear term above.

Hamiltonian Systems and the Symplectic Structure

The Symplectic Form

Definition. A symplectic manifold $(M,\omega)$ is a smooth manifold with a closed nondegenerate $2$-form $\omega$. For a function $H : M\to\mathbb{R}$, the Hamiltonian vector field $X_H$ is defined by

$$ \iota_{X_H}\omega = dH , $$

and the Poisson bracket of $f,g\in C^\infty(M)$ is

$$ \{f,g\} = \omega(X_f,X_g) . $$

Theorem (properties of the bracket). The bracket is bilinear and skew-symmetric, satisfies the Leibniz rule $\{f,gh\}=\{f,g\}h+g\{f,h\}$, and satisfies the Jacobi identity

$$ \{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0 ; $$

consequently $C^\infty(M)$ is a Lie algebra under the bracket and the map $f\mapsto X_f$ is a Lie algebra homomorphism, $[X_f,X_g]=X_{\{f,g\}}$. The bracket is nondegenerate in the sense that $\{f,g\}=0$ for all $g$ implies that $f$ is locally constant.

Proof. The bilinearity, skew-symmetry and Leibniz rule follow from the corresponding properties of $\omega$ and of the differential; the Jacobi identity is the identity $d\omega=0$ written in terms of the bracket, and the homomorphism property is the computation $[X_f,X_g]=\omega$-dual of $d\{f,g\}$. The details are those of Symplectic Forms and Poisson Brackets and of the symplectic geometry of Part II.

Theorem (Darboux). Every symplectic manifold is locally symplectomorphic to $(\mathbb{R}^{2n},\sum_{i=1}^n dq^i\wedge dp_i)$: around every point there are coordinates $(q^1,\dots,q^n,p_1,\dots,p_n)$, the canonical coordinates, in which the form is the displayed one.

Proof. Quoted as standard and belonging to the symplectic geometry of Part II, written in parallel; the proof is a Moser-type argument using the closedness and nondegeneracy of $\omega$ and a homotopy of forms.

Corollary (Hamilton's equations in canonical coordinates). In canonical coordinates, $X_H$ has components

$$ \dot q^i = \frac{\partial H}{\partial p_i}, \qquad \dot p_i = -\frac{\partial H}{\partial q^i} , $$

and the Poisson bracket is

$$ \{f,g\} = \sum_{i=1}^n\left(\frac{\partial f}{\partial q^i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q^i}\right) . $$

Moreover $\frac{d}{dt}f = \{f,H\}$ along the flow of $X_H$, so that $f$ is conserved exactly when $\{f,H\}=0$.

Proof. The form $\omega = \sum dq^i\wedge dp_i$ gives $\omega(X_H,Y)=dq(X_H)\,dp(Y)-dp(X_H)\,dq(Y) = \dot q\,dp(Y)+\dot p\,dq(Y)$ for $X_H=(\dot q,\dot p)$; requiring this to equal $dH(Y)=H_qdq(Y)+H_pdp(Y)$ for all $Y$ gives the equations. The bracket formula is the same computation applied to $f$ and $g$, and the last statement is $\frac{d}{dt}f = df(X_H)=X_H(f)=\{f,H\}$.

Definition. A Poisson manifold is a manifold with a bracket on its functions that is bilinear, skew-symmetric, satisfies the Leibniz rule and the Jacobi identity; the bracket need not come from a symplectic form, and its rank may drop on a subvariety. A Poisson structure is the corresponding section of $\Lambda^2TM$; when it is nondegenerate it is the inverse of a symplectic form.

Remark (the two formulations). A symplectic structure is a nondegenerate Poisson structure, and the classical mechanics of a system without constraints is symplectic; a Poisson structure that degenerates occurs for a constrained system, after symmetry reduction, and for the Lie–Poisson structure on the dual of a Lie algebra. The relation between the two is the content of the symplectic geometry of Part II, where the reduction of a symplectic manifold by a group action is treated; here the nondegenerate case is used, and the degenerate case is named only where it occurs, as in the Euler top below.

The Bracket Formulation of the Dynamics and Poisson's Theorem

The bracket turns the differential equations of the flow into an algebraic statement about functions on phase space.

Theorem (the bracket formulation). Along the flow of $X_H$, a smooth function $g(q,p,t)$ satisfies

$$ \frac{d}{dt}g = \{g,H\} + \frac{\partial g}{\partial t} , $$

so that $g$ is a constant of the motion if and only if $\{g,H\}+\partial_tg=0$; when $g$ has no explicit time dependence the condition is $\{g,H\}=0$.

Proof. The derivative of $g$ along a curve of the flow has one part from the motion in phase space and one part from the explicit time dependence, $\frac{d}{dt}g=dg(X_H)+\partial_tg$, and $dg(X_H)=X_H(g)=\{g,H\}$ by the definition of the bracket. ∎

The bracket formulation is the reason the bracket, and not the vector field, is the working object of Hamiltonian mechanics: a conserved quantity is a function whose bracket with the Hamiltonian vanishes, and the search for integrals is the search for the centraliser of $H$.

Theorem (Poisson). If $u$ and $v$ are constants of the motion of an autonomous Hamiltonian $H$, then their bracket $\{u,v\}$ is a constant of the motion.

Proof. Along the flow, $\frac{d}{dt}\{u,v\}=\{\{u,v\},H\}$ by the bracket formulation. The identity $$ \{\{u,v\},H\}=\{\{u,H\},v\}+\{u,\{v,H\}\} , $$ which is the Jacobi identity together with the Leibniz rule, says that the map $g\mapsto\{g,H\}$ is a derivation; both terms on the right vanish because $\{u,H\}=0$ and $\{v,H\}=0$. Hence $\{u,v\}$ is conserved. ∎

Corollary (the constants of motion form a Lie algebra). The constants of the motion of $H$ are closed under the bracket, so they form a Lie subalgebra of $C^\infty(M)$, the centraliser of $H$. The bracket of two known integrals is a further integral, which may be a function of the integrals already known — as the three components of the angular momentum close on one another — or may be independent and enlarge the algebra.

Remark (the reach and the limit of the theorem). Poisson's theorem manufactures a new constant of motion from two old ones and closes the family of integrals under the bracket, and it needs no symmetry and no hypothesis beyond the conservation of its two inputs. It is not, however, a machine for integrability. The bracket it produces may vanish, or may merely reproduce the integrals already known, or may generate a finite-dimensional algebra of dimension smaller than the number of degrees of freedom; and for a generic Hamiltonian the only independent integral is typically the energy itself, on which the theorem has nothing to act. The Kepler problem is the classical case in which the algebra closes, the brackets of the angular momentum and the Runge–Lenz vector reproducing the finite algebra of the integrals, while a generic central potential has only the angular momentum and the energy.

Canonical Transformations and Generating Functions

The symplectic form fixes the flow but not the coordinates: a change of phase-space variables that pulls the form back to itself preserves the form of Hamilton's equations, and this freedom is what allows a system to be solved by passing to coordinates in which it is trivial. The transformations of this kind are the canonical transformations, and the Hamilton–Jacobi equation of the calculus of variations produces the one that solves the system, as the companion article The Hamilton–Jacobi Equation and Separability develops.

Definition. A canonical transformation, or symplectomorphism, of a symplectic manifold $(M,\omega)$ is a diffeomorphism $\varphi$ with $\varphi^*\omega=\omega$; a local canonical transformation is a diffeomorphism between open sets with the same property.

Theorem (the criteria). Let $\varphi : (q,p)\mapsto(Q,P)$ be a diffeomorphism of a coordinate neighbourhood of $(\mathbb{R}^{2n},\sum_i dq^i\wedge dp_i)$. The following are equivalent:

the form is preserved, $\sum_i dQ^i\wedge dP_i=\sum_i dq^i\wedge dp_i$;
the Poisson bracket is preserved, $\{\varphi^*f,\varphi^*g\}=\varphi^*\{f,g\}$ for all smooth $f,g$;
the canonical brackets of the images hold, $\{Q^i,Q^j\}=0$, $\{P_i,P_j\}=0$ and $\{Q^i,P_j\}=\delta^i_j$, the bracket being evaluated in the original variables;
every Hamiltonian flow is carried to the flow of the pulled-back Hamiltonian, so that Hamilton's equations keep their form;
the Jacobian is symplectic, $(D\varphi)^{\mathsf T}\Omega(D\varphi)=\Omega$, with $\Omega$ the matrix of $\omega$, $\Omega_{i,n+j}=\delta_{ij}$ and $\Omega_{n+i,j}=-\delta_{ij}$.

Proof sketch. Preservation of $\omega$ is the first clause, and it is equivalent to the preservation of the bracket because the bracket is $\omega(X_f,X_g)$ and the nondegeneracy of $\omega$ lets the brackets of the coordinate functions determine the form. A bracket-preserving map carries $X_H$ to $X_{\varphi^*H}$ and hence preserves Hamilton's equations, and conversely; and the matrix clause is the first clause differentiated at a point. ∎

Theorem (generating functions). Let $\varphi$ be a local canonical transformation whose graph is simply connected, and let $F_1$ be a function of the old and the new coordinates with

$$ p_i = \frac{\partial F_1}{\partial q^i}, \qquad P_i = -\frac{\partial F_1}{\partial Q^i} . $$

Then $\varphi$ is canonical; and every such transformation has a generating function $F_1$, unique up to an additive constant. Replacing $F_1$ by its Legendre transform in the variables that are exchanged gives the three other types, one for each choice of one old and one new coordinate per degree of freedom:

$$ F_2(q,P) = F_1 + \sum_iQ^iP_i, \qquad p_i = \frac{\partial F_2}{\partial q^i}, \qquad Q^i = \frac{\partial F_2}{\partial P_i} ; $$

$$ F_3(p,Q) = F_1 - \sum_iq^ip_i, \qquad q^i = -\frac{\partial F_3}{\partial p_i}, \qquad P_i = -\frac{\partial F_3}{\partial Q^i} ; $$

$$ F_4(p,P) = F_1 + \sum_iQ^iP_i - \sum_iq^ip_i, \qquad q^i = -\frac{\partial F_4}{\partial p_i}, \qquad Q^i = \frac{\partial F_4}{\partial P_i} . $$

Proof sketch. The graph of a canonical transformation carries the two symplectic potentials $\sum_i p_idq^i$ and $\sum_i P_idQ^i$, and their difference is closed because both differentiate to $\omega$ on the graph; the graph being simply connected, the difference is exact, $\sum_i p_idq^i-\sum_i P_idQ^i=dF_1$, and reading the differential in the independent variables $(q,Q)$ gives the two relations. The Legendre transforms eliminate the variables that are no longer independent, the invertibility of each transform being the regularity of the canonical map in that pair. ∎

Theorem (time-dependent generating functions). If the generating function depends on the time, the transformation is canonical at each fixed $t$ and the new Hamiltonian is

$$ K = H + \frac{\partial F}{\partial t} , $$

for any of the four types $F$.

Proof. Extend the exactness relation to include the time: on the graph the one-form $\sum_i p_idq^i-\sum_i P_idQ^i+(K-H)dt$ is exact with potential $F$, which is the statement $p=F_q$, $P=-F_Q$ and $K=H+F_t$ when $F$ is read in $(q,Q,t)$; the Legendre transforms carry the computation to the other types. ∎

The Hamiltonian is therefore fixed only up to the partial time derivative of an arbitrary function of the coordinates, the momenta and the time, which is the ambiguity that the Hamiltonian carries in every formulation; requiring the derivative to vanish is what produces the Hamilton–Jacobi equation below.

Examples. The elementary transformations, each of which satisfies the criteria and is worth reading against its generating function:

The identity is generated by $F_2=\sum_iq^iP_i$, since $p_i=\partial F_2/\partial q^i=P_i$ and $Q^i=\partial F_2/\partial P_i=q^i$.
The coordinate scaling is generated by $F_2=\sum_i\lambda_iq^iP_i$ with $\lambda_i>0$, giving $Q^i=\lambda_iq^i$ and $P_i=p_i/\lambda_i$; the two scalings are reciprocal, which is why the bracket is preserved.
The swap is generated by $F_1=\sum_iq^iQ^i$, giving $p_i=Q^i$ and $P_i=-q^i$, so that the coordinates and the momenta exchange roles with a sign.
The shear of the momenta is generated by $F_2=\sum_iq^iP_i+\tfrac12\sum_ia_i(q^i)^2$, giving $Q^i=q^i$ and $P_i=p_i-a_iq^i$; the generating function is quadratic, and the transformation is the elementary one that the linear canonical transformations are built from.
For the one-dimensional oscillator the rotation $(Q,P)=(q\cos\tau-p\sin\tau,\,q\sin\tau+p\cos\tau)$ is canonical for every $\tau$, and it is the flow of the oscillator after the time $\tau$; the bracket $\{Q,P\}=1$ is the statement that the flow preserves the form.

Remark (the Hamilton–Jacobi equation). Ask that the new Hamiltonian vanish, $K=H+\partial F/\partial t=0$, and take the generating function to be $F_2=S(q,P,t)$. The first relation, $p_i=\partial S/\partial q^i$, turns the condition into

$$ \frac{\partial S}{\partial t}+H\Bigl(q,\frac{\partial S}{\partial q},t\Bigr)=0 , $$

which is the Hamilton–Jacobi equation of the calculus of variations. The action, read as a function of the coordinates and the new momenta, is the generating function of the canonical transformation that trivialises the flow: the new coordinates $Q^i=\partial S/\partial P_i$ are constant along the motion, and in the integrable case the transformation produced in this way is the passage to the action-angle variables. The equation, its complete integral, the separation of variables and the Stäckel conditions are developed in the companion article The Hamilton–Jacobi Equation and Separability. The promise of the calculus of variations, that the Hamiltonian formalism, its canonical transformations and its Poisson brackets belong to this article, is discharged by the two theorems above.

Remark (the group of canonical transformations). The canonical transformations of $(M,\omega)$ form a group under composition, and its identity component is generated by the Hamiltonian flows: a one-parameter family $\varphi_t$ of canonical transformations with $\varphi_0=\mathrm{id}$ is the flow of a Hamiltonian vector field, and the Poisson bracket is its Lie algebra, since $f\mapsto X_f$ is a Lie algebra homomorphism. The symplectic geometry of Part II is the study of this group and of the manifolds on which it acts.

Remark (infinitesimal canonical transformations and the generator of the flow). A one-parameter family $\varphi_\varepsilon$ of canonical transformations with $\varphi_0=\mathrm{id}$ moves every phase-space function by $\delta\eta=\varepsilon\{\eta,\sigma\}$ to first order in $\varepsilon$, for a single function $\sigma$, the generator of the family; conversely every smooth $\sigma$ defines such a family, and since $\{\eta,\sigma\}=d\eta(X_\sigma)$ the finite transformation is the flow of $X_\sigma$ run for the parameter. Two readings of the generator fix its role.

The generator $\sigma=H$ gives $\delta q^i=\varepsilon\{q^i,H\}=\varepsilon\,\partial H/\partial p_i$ and $\delta p_i=\varepsilon\{p_i,H\}=-\varepsilon\,\partial H/\partial q^i$; the infinitesimal canonical transformation generated by the Hamiltonian is exactly the infinitesimal time evolution, so the Hamiltonian generates the flow, and the finite flow is the exponential of the generator.
A generator that is conserved, $\{\sigma,H\}=0$, produces a family leaving $H$ invariant, since $\delta_\sigma H=\varepsilon\{H,\sigma\}=0$; conversely, invariance of $H$ under the family is the conservation of its generator. This is the infinitesimal form of the reciprocity between symmetry and conservation, and it reproduces the momentum map: the generator of a translation is the conjugate momentum, since $\{q,p\}=1$, and the generator of a rotation is the corresponding component of the angular momentum, since $\{x,L_z\}=-y$ and $\{y,L_z\}=x$ are the infinitesimal rotation.

Conservation Laws and Phase Volume

Theorem (conservation of energy). For an autonomous Hamiltonian $H$, the function $H$ is conserved by its own flow, $\{H,H\}=0$, and the flow preserves the symplectic form, $\mathcal{L}_{X_H}\omega=0$, and hence the volume form $\omega^n/n!$.

Proof. The bracket is skew, so $\{H,H\}=0$; the preservation of $\omega$ is Cartan's formula $\mathcal{L}_{X_H}\omega = d\iota_{X_H}\omega+\iota_{X_H}d\omega = d\,dH + 0 = 0$, using closedness. The volume form is a power of $\omega$, so it is preserved as well.

Theorem (Liouville). The flow of a Hamiltonian vector field on a symplectic manifold of dimension $2n$ preserves the measure induced by $\omega^n/n!$; consequently, on a bounded invariant set of finite measure, almost every orbit returns arbitrarily close to its starting point infinitely often (Poincaré recurrence).

Proof. The invariance of the volume form is the preceding theorem. For the recurrence, let $A$ be the invariant set of finite positive measure and let $U\subseteq A$ be open; the images $\phi_t(U)$ all have the same measure as $U$, so if they were pairwise disjoint their union would have infinite measure; hence $\phi_{t_1}(U)\cap\phi_{t_2}(U)\neq\emptyset$ for some $t_1

Remark (Liouville's equation and the density). The preservation of volume has a local form. Let $\rho(q,p,t)$ be a density and transport the measure $\rho\,\omega^n/n!$ with the flow; then $\rho$ satisfies the continuity equation

$$ \frac{\partial\rho}{\partial t}+\operatorname{div}(\rho X_H)=0 . $$

The flow is divergence-free, $\operatorname{div}X_H=0$, precisely because it preserves the volume form, so the continuity equation is the Liouville equation

$$ \frac{\partial\rho}{\partial t}+\{\rho,H\}=0 . $$

A distribution of initial conditions is therefore transported by the flow exactly when it satisfies this equation, and its stationary solutions, those with $\partial_t\rho=0$, are the densities conserved by the flow. Every function of the Hamiltonian is one, since $\{f(H),H\}=f'(H)\{H,H\}=0$, in particular the Gibbs density $e^{-\beta H}/Z$ of statistical mechanics; the volume form itself is the case $\rho$ constant, and the equation is the differential form of the incompressibility of the flow.

Theorem (Noether, Hamiltonian form). Let a Lie group $G$ act on the symplectic manifold $(M,\omega)$ preserving $\omega$ and the Hamiltonian $H$, with infinitesimal generators $\xi_{\mathrm{G}}$ and a momentum map $J : M\to\mathrm{G}^*$ satisfying $d\langle J,\xi\rangle = \iota_{\xi_M}\omega$. Then $J$ is conserved along the flow of $H$: $\{J_\xi,H\}=0$ for every $\xi\in\mathrm{G}$.

Proof. The invariance of $H$ under the group gives $\mathcal{L}_{\xi_M}H = 0$, i.e., $dH(\xi_M)=0$. On the other hand $dH(\xi_M)=\omega(X_H,\xi_M)$ by the definition of $X_H$, and $\omega(X_H,\xi_M)=-\omega(\xi_M,X_H)=-d\langle J,\xi\rangle(X_H)=-\{J_\xi,H\}$, because $X_{J_\xi}=\xi_M$ by the definition of the momentum map and the nondegeneracy of $\omega$. Hence the bracket vanishes. The one-parameter case is the Noether theorem of the calculus of variations, from which the statement descends.

Complete Integrability and Action-Angle Variables

Definition. A Hamiltonian system $(M,\omega,H)$ with $\dim M=2n$ is completely integrable (in the sense of Liouville) if there exist $n$ functions $F_1=H,F_2,\dots,F_n$ on $M$ that are independent at generic points and pairwise in involution, $\{F_i,F_j\}=0$ for all $i,j$.

Theorem (Liouville–Arnold). Let $(M,\omega,H)$ be completely integrable and let $c$ be a regular value of $F=(F_1,\dots,F_n)$ such that the level set $M_c=F^{-1}(c)$ is compact and connected. Then:

$M_c$ is diffeomorphic to the $n$-torus $\mathbb{T}^n$;
the flow of $X_H$ is linear on $M_c$ in suitable coordinates, so the motion is conditionally periodic with $n$ frequencies;
there is a neighbourhood of $M_c$ with action-angle coordinates $(I,\theta)\in\mathbb{R}^n\times\mathbb{T}^n$ in which $\omega = \sum d\theta^i\wedge dI_i$ and $H = H(I)$ depends only on the actions.

Proof. Quoted as standard. The vector fields $X_{F_i}$ are tangent to $M_c$ because the $F_i$ are in involution; they commute, since $[X_{F_i},X_{F_j}]=X_{\{F_i,F_j\}}=0$, so they define an integrable distribution whose leaves are open subsets of $M_c$. Compactness and connectedness make each leaf a torus, giving the first assertion. The leaves are the orbits of an abelian group of translations, and the parameters along the commuting flows are the angles, giving the second. The actions are defined by integrating the $1$-forms $\iota_{X_{F_i}}\omega$ over a basis of cycles of the torus; that the integrals are locally constant in $c$ follows from the closedness of those forms, and the resulting coordinates are canonical because the cycles are Lagrangian, which gives the third assertion.

Corollary (solution by quadrature). In action-angle coordinates the equations of the flow are $\dot I=0$ and $\dot\theta = \omega(I) = \frac{\partial H}{\partial I}$, so $I$ is constant and $\theta(t)=\theta(0)+\omega(I)t$; the system is solved by a single integration of known functions.

Remark (the relation to the integrable equations). The Liouville–Arnold theorem is the finite-dimensional half of the theory of integrable systems. The travelling-wave reductions of the KdV, nonlinear Schrödinger and sine-Gordon equations — obtained by substituting a travelling-wave ansatz into the partial differential equation — are finite-dimensional Hamiltonian systems, and their integrability, where it holds, is the integrability of this theorem; the infinite-dimensional Lax pairs of the two preceding articles reduce to finite-dimensional ones on these invariant submanifolds. The theorem also delimits integrability: a generic perturbation of an integrable system destroys the invariant tori, and the Kolmogorov–Arnold–Moser theorem describes the tori of a small perturbation that survive, those whose frequency vectors are sufficiently nonresonant; the whole subject is a chapter of dynamical systems, and the articles on that subject in this Part, written in parallel, treat the stability and the destruction of the tori.

Examples

Example (the Kepler problem). For a particle of unit mass in a central potential $-k/r$ in the plane, the configuration space is $\mathbb{R}^2\setminus\{0\}$ with polar coordinates, $H = \frac12(p_r^2+p_\theta^2/r^2)-k/r$, and the system is completely integrable with the commuting pair $F_1=H$ and $F_2=p_\theta$, the angular momentum. The level sets of $(H,p_\theta)$ are tori except for the separatrix of the parabolic orbit, the actions are the classical Delaunay variables, and the additional conserved vector (the Runge–Lenz vector) accounts for the further degeneracy of the frequency vector: the two frequencies of the bounded motion coincide, so the bounded orbits close and are ellipses rather than dense on a torus. The example shows that a system may have more integrals than the Liouville theorem requires, and that the extra integral is detected by a resonance of the frequency vector.

Example (the Euler top). The rigid body with a fixed point and no external torque has phase space the dual of the Lie algebra $\mathrm{SO}(3)$, with the Lie–Poisson structure and the Hamiltonian $H = \frac12\bigl(M_1^2/I_1+M_2^2/I_2+M_3^2/I_3\bigr)$; equivalently, in the body frame, the Euler equations $\dot M = M\times(I^{-1}M)$. The Casimir function of the Lie–Poisson structure, $\tfrac12(M_1^2+M_2^2+M_3^2)$, is conserved, and together with the energy and the component of the angular momentum along a fixed axis of space it gives three commuting integrals in the six-dimensional phase space $T^*SO(3)$, so the system is completely integrable, and the reduced level sets are the classical ellipsoids cut by the energy spheres; the free symmetric top has a further degeneracy and the motion is a regular precession. The example is the standard instance of a system whose integrability is read off a Poisson structure that is not symplectic, and whose reduction is the setting of the symplectic geometry of Part II. The reduction itself — from the cotangent bundle $T^*SO(3)$ to the dual of the Lie algebra, the Lie–Poisson equation it carries, and the variational Euler–Poincaré equation that yields the same system — is treated in the companion article Lie–Poisson Reduction and the Euler–Poincaré Equation, which also derives the bracket of Poisson Geometry and carries the construction to the volume-preserving diffeomorphism group, where the equation of the top becomes the Euler equation of an ideal incompressible flow.

Example (the geodesic flow). For the mechanical Lagrangian with zero potential, the Hamiltonian flow on $T^*Q$ is the geodesic flow of the metric $g$. It is the model of a Hamiltonian system whose dynamics is chaotic for a negatively curved metric — the flow on a compact quotient of the hyperbolic plane is Anosov and ergodic, as the articles of this Part on the geodesic flow and on hyperbolic dynamics record — and completely integrable for a metric with sufficiently many Killing fields. The comparison of the two cases is the classical instance of the dichotomy between integrable and chaotic Hamiltonian dynamics.

Summary

A Lagrangian system is given by a function $L$ on the tangent bundle of a configuration space $Q$, and its motions are the critical points of the action $S[q]=\int L(q,\dot q,t)dt$, equivalently the solutions of the Euler–Lagrange equations; the momentum $p_i=\partial L/\partial\dot q^i$ defines the Legendre transform, and when the transform is invertible the system is equivalently a Hamiltonian system on the cotangent bundle with $H=\sum p_i\dot q^i-L$ and Hamilton's equations $\dot q=\partial H/\partial p$, $\dot p=-\partial H/\partial q$. The Hamiltonian formulation is the symplectic one: a symplectic manifold $(M,\omega)$, the Hamiltonian vector field defined by $\iota_{X_H}\omega=dH$, the Poisson bracket $\{f,g\}=\omega(X_f,X_g)$ with its Jacobi identity and Leibniz rule, and the local model of Darboux $\omega=\sum dq^i\wedge dp_i$; the flow preserves $\omega$ and the volume $\omega^n/n!$, which gives Liouville's theorem and Poincaré recurrence, and a symmetry of the Hamiltonian gives a conserved momentum map by Noether's theorem. A Hamiltonian system on a $2n$-dimensional manifold is completely integrable when it possesses $n$ independent integrals in involution; then, on a compact connected regular level set, the Liouville–Arnold theorem exhibits an invariant torus with linear (conditionally periodic) flow and action-angle coordinates in which $H$ depends only on the actions, so the system is solved by quadrature. The Euler top, the Kepler problem and the geodesic flow are the standard examples, and the theory is the finite-dimensional companion of the integrable systems of the preceding articles, whose travelling-wave reductions are systems of this kind. A coordinate on which the Lagrangian does not depend is cyclic, and its conjugate momentum is then conserved. The equation of a regular system can also be written without coordinates: the canonical momentum one-form $\theta_L=p_i\,dq^i$ is the pullback of the tautological form by the Legendre transform, the field of the system is the unique second-order vector field $X$ with $\mathcal{L}_X\theta_L=dL$, and the integral curves of $X$ are the velocity lifts of the solutions.

The mechanical origin of the equation is D'Alembert's principle, the statement that the impressed forces and the inertial terms do equal virtual work across every displacement consistent with the constraints; on generalised coordinates it becomes $\frac{d}{dt}T_{\dot q^j}-T_{q^j}=Q_j$, and for conservative forces the Euler–Lagrange equation of $L=T-V$. A constraint of the form $\phi(q,t)=0$ is holonomic and is used by elimination; a velocity constraint $A^a_j(q)\dot q^j=0$ with no integrating factor is non-holonomic, and it is imposed by multipliers in the Lagrange–d'Alembert equations $\frac{d}{dt}L_{\dot q^j}-L_{q^j}=\lambda_aA^a_j$, which are not the stationarity of an action but the transcription of D'Alembert's principle. The energy is still conserved for an autonomous Lagrangian and a velocity-homogeneous constraint, because the multiplier term is annihilated by the constraint; and the multiplier rule is the finite-dimensional form of the costate of optimal control.

A change of phase-space variables preserving the symplectic form is a canonical transformation, and it is characterised equally by the preservation of the Poisson bracket, by the canonical brackets $\{Q^i,Q^j\}=\{P_i,P_j\}=0$, $\{Q^i,P_j\}=\delta^i_j$ of the images, by the preservation of the form of Hamilton's equations, and by a symplectic Jacobian. Locally such a transformation is generated by a function of one old and one new coordinate per degree of freedom, in one of the four types $F_1(q,Q)$, $F_2(q,P)$, $F_3(p,Q)$, $F_4(p,P)$ related by Legendre transforms, with $p=F_q$ and $P=-F_Q$ for the first; the identity, the coordinate scaling, the exchange of coordinates and momenta and the quadratic shear of the momenta are the elementary instances. A generating function depending on the time shifts the Hamiltonian by its partial derivative, $K=H+F_t$; requiring the new Hamiltonian to vanish turns this into the Hamilton–Jacobi equation of the calculus of variations, and the action read as a function of the coordinates and the new momenta is then the generating function that trivialises the flow. The canonical transformations form a group whose identity component is generated by the Hamiltonian flows, and the Poisson bracket is its Lie algebra.

Along the flow of $X_H$ a function evolves by $\frac{d}{dt}g=\{g,H\}+\partial_tg$, so the constants of the motion are the functions with $\{g,H\}=0$, and the bracket of two of them is a third by Poisson's theorem: the constants of the motion form the Lie subalgebra that centralises $H$. The Hamiltonian equations are also the Euler–Lagrange equations of the modified Lagrangian $\tilde L=\sum_ip_i\dot q^i-H$, varied in $q$ and in $p$ independently, and they are the flow of the infinitesimal canonical transformation generated by $H$: a generator $\sigma$ displaces every function by $\varepsilon\{\cdot,\sigma\}$, and $\sigma$ is conserved exactly when it leaves $H$ invariant, which is the infinitesimal reciprocity of symmetry and conservation and reproduces the momentum map. The preservation of phase volume has the local form of Liouville's equation $\partial_t\rho+\{\rho,H\}=0$ for a density, whose stationary solutions are the conserved densities, every function of the Hamiltonian among them. In one degree of freedom the level curves of $H$ are the phase portrait, with centres at the definite critical points and saddles at the indefinite ones, and the separatrix through a saddle separates libration from rotation, as the pendulum illustrates.

Summary of Notation

Symbol	Meaning
$Q$, $TQ$, $T^*Q$	Configuration space, tangent and cotangent bundles
$L(q,\dot q,t)$	Lagrangian; $S[q]$ the action
Euler–Lagrange	$\frac{d}{dt}L_{\dot q^i}-L_{q^i}=0$
$p_i$, $\mathbb{F}L$	Momentum and Legendre transform
cyclic $q^k$	$\partial L/\partial q^k=0$; $p_k$ conserved
$\theta_L=p_i\,dq^i$	Canonical momentum one-form of $L$
$\mathcal{L}_X\theta_L=dL$	Coordinate-free Euler–Lagrange equation
$\delta q$, $Q_j$	Virtual displacement and generalised force
$A^a_j$, $\lambda_a$	Non-holonomic constraint matrix and its multipliers
$H(q,p)$	Hamiltonian, $\sum p_i\dot q^i-L$
Hamilton's equations	$\dot q^i=H_{p_i}$, $\dot p_i=-H_{q^i}$
$(M,\omega)$	Symplectic manifold, closed nondegenerate $2$-form
$X_H$, $\iota_{X_H}\omega=dH$	Hamiltonian vector field
$\{f,g\}$	Poisson bracket, $\omega(X_f,X_g)$
$\frac{d}{dt}g=\{g,H\}+\partial_tg$	Evolution of a function along the Hamiltonian flow
Poisson's theorem	The bracket of two constants of motion is a constant of motion
$\tilde L=\sum_ip_i\dot q^i-H$	Modified Lagrangian of the independent-variation principle
$\sigma$, $\delta\eta=\varepsilon\{\eta,\sigma\}$	Generator of an infinitesimal canonical transformation
$\partial_t\rho+\{\rho,H\}=0$	Liouville's equation for a density
separatrix	Level curve through a hyperbolic (saddle) equilibrium
$\{Q^i,P_j\}=\delta^i_j$	Canonical brackets of a canonical transformation
$F_1$, $F_2$, $F_3$, $F_4$	Generating functions of a canonical transformation
$K=H+\partial F/\partial t$	Hamiltonian in the new coordinates
$g_{ij}$, $g^{ij}$	Metric and inverse metric of $Q$
momentum map $J$	Conserved quantity associated with a symmetry
$F_1,\dots,F_n$	Integrals in involution, $F_1=H$
$(I,\theta)$	Action-angle coordinates, $\omega=\sum d\theta^i\wedge dI_i$
$\omega(I)$	Frequency vector $\partial H/\partial I$ of the torus flow
KAM	Kolmogorov–Arnold–Moser survival of tori under perturbation