1 · basic mechanics

newtonian foundations

Empirical observation shows that the motion of a free particle is uniform and rectilinear in an inertial frame, and that the acceleration of a particle is proportional to the applied force. Space-time is modeled as an affine four-dimensional manifold equipped with a Galilean structure: a preferred absolute time function $t$ whose level sets are Euclidean three-spaces, together with a Euclidean metric on each spatial slice. The Galilean group of transformations that preserve this structure consists of spatial rotations, translations, and uniform boosts.

definition (inertial frame). a chart of Galilean space-time in which free particles have vanishing acceleration, so that uniform rectilinear motion is the default law of free motion.

definition (newton's principle of determinacy). the state of a mechanical system at any instant is completely fixed by the positions and velocities of all particles at that instant. Consequently the equations of motion are second-order ordinary differential equations on configuration space.

For a system of $N$ particles the configuration space is $\mathbb{R}^{3N}$ (or a submanifold thereof when constraints are present), and the Newtonian equations read $$ m_i\ddot{\mathbf{q}}_i=\mathbf{F}_i(\mathbf{q},\dot{\mathbf{q}},t),\qquad i=1,\dots,N. $$ The mathematical motivation for the affine-Galilean model is that it is the unique (up to isomorphism) structure compatible with the observed invariance under the Galilean group and with the absolute character of time intervals.

remark (coordinate dependence). as written, the components $\mathbf{F}_i$ and $\mathbf{q}_i$ depend on the choice of inertial coordinates. The modern program replaces this chart-based statement with tensor and manifold language so that the law of motion itself becomes independent of coordinates while recovering Newton's form in any local inertial frame.

low dimension and central forces

For a single particle in a conservative force field $\mathbf{F}=-\nabla V$ the Newtonian equation reduces to $$ m\ddot{\mathbf{q}}=-\nabla V(\mathbf{q}). $$ When the force is central, $V=V(r)$ with $r=|\mathbf{q}|$, angular momentum is conserved and the plane of motion is fixed.

theorem (conservation of angular momentum for central forces). let $m\ddot{\mathbf{q}}=-\nabla V(|\mathbf{q}|)$ with $V$ of class $C^1$ on $\mathbb{R}^3\setminus\{0\}$. then $$ \mathbf{L}:=\mathbf{q}\times m\dot{\mathbf{q}} $$ is constant along every solution.

proof. differentiate: $$ \frac{d\mathbf{L}}{dt} =\dot{\mathbf{q}}\times m\dot{\mathbf{q}} +\mathbf{q}\times m\ddot{\mathbf{q}} =\mathbf{0} +\mathbf{q}\times\bigl(-\nabla V(r)\bigr). $$ for $r=|\mathbf{q}|$ one has $\nabla V(r)=V'(r)\,\mathbf{q}/r$, so $\mathbf{q}\times\nabla V(r)=\mathbf{0}$. hence $d\mathbf{L}/dt=\mathbf{0}$. $\square$

corollary (planar motion and effective potential). if $\mathbf{L}\neq\mathbf{0}$, the motion lies in the plane orthogonal to $\mathbf{L}$ through the origin. writing polar coordinates $(r,\theta)$ in that plane with conserved $\ell=|\mathbf{L}|$, the radial equation is that of a one-dimensional particle with energy $$ \frac12 m\dot r^2+V_{\mathrm{eff}}(r)=E,\qquad V_{\mathrm{eff}}(r)=V(r)+\frac{\ell^2}{2mr^2}. $$ the phase portrait of the radial dynamics is therefore completely determined by the critical points and level sets of $V_{\mathrm{eff}}$.

proof (reduction). from $\mathbf{L}$ constant one may choose coordinates so that $\mathbf{L}=(0,0,\ell)$ with $\ell=mr^2\dot\theta$. energy conservation $E=\frac12 m|\dot{\mathbf{q}}|^2+V(r)$ together with $|\dot{\mathbf{q}}|^2=\dot r^2+r^2\dot\theta^2$ and $\dot\theta=\ell/(mr^2)$ yields the displayed $V_{\mathrm{eff}}$. $\square$

The Kepler problem $V(r)=-k/r$ admits the explicit solution $$ r=\frac{p}{1+e\cos\theta}, $$ where $p=\ell^2/(mk)$ is the semi-latus rectum and $e$ is the eccentricity. The orbit is a conic section with one focus at the origin. The geometric origin of this fact is the isotropy of Euclidean space: the only rotationally invariant force laws that produce closed bounded orbits for all nearby initial data of a given type (Bertrand's theorem) are the harmonic oscillator and the inverse-square law; the latter yields the focal property of the ellipse.

remark (bertrand). the full classification theorem of Bertrand is classical but lengthy; its role here is conceptual: rotational symmetry alone severely restricts which central potentials give permanently closed bounded orbits, and only two local force laws survive.

lagrangian formalism

Hamilton's principle states that the actual trajectory $q(t)$ between fixed end-points realizes a critical point of the action functional $$ S[q]=\int_{t_0}^{t_1}L(q,\dot{q},t)\,dt. $$ Configuration space $Q$ is taken to be a smooth manifold. The Lagrangian becomes a smooth function $L:TQ\to\mathbb{R}$ on the tangent bundle. The variational principle depends only on the values of $L$ along curves, not on any particular coordinate chart.

definition (action of a path). for $q\in C^2([t_0,t_1],Q)$ with prescribed endpoints $q(t_0)=q_0$, $q(t_1)=q_1$, $$ S[q]=\int_{t_0}^{t_1}L\bigl(q(t),\dot{q}(t),t\bigr)\,dt, $$ where $(q,\dot q)$ denotes the velocity curve in $TQ$.

theorem (euler-lagrange equations). if $q$ is a critical point of $S$ among variation of paths with fixed endpoints, then in every local coordinate chart on $Q$, $$ \frac{d}{dt}\Bigl(\frac{\partial L}{\partial\dot{q}^i}\Bigr) -\frac{\partial L}{\partial q^i}=0,\qquad i=1,\dots,\dim Q. $$ these equations transform covariantly under $C^2$ changes of coordinates on $Q$.

proof. work in a chart and write $q(t)=(q^1(t),\dots,q^n(t))$. let $\eta$ be a $C^2$ vector field along $q$ with $\eta(t_0)=\eta(t_1)=0$, and set $q_\varepsilon=q+\varepsilon\eta$ for small $\varepsilon$. then $$ \frac{d}{d\varepsilon}\Big|_{\varepsilon=0}S[q_\varepsilon] =\int_{t_0}^{t_1}\Bigl( \frac{\partial L}{\partial q^i}\eta^i +\frac{\partial L}{\partial\dot{q}^i}\dot\eta^i \Bigr)\,dt. $$ integrate the second summand by parts: $$ \int_{t_0}^{t_1}\frac{\partial L}{\partial\dot{q}^i}\dot\eta^i\,dt =\Bigl[\frac{\partial L}{\partial\dot{q}^i}\eta^i\Bigr]_{t_0}^{t_1} -\int_{t_0}^{t_1}\frac{d}{dt}\Bigl(\frac{\partial L}{\partial\dot{q}^i}\Bigr)\eta^i\,dt. $$ the boundary term vanishes because $\eta(t_0)=\eta(t_1)=0$. criticality therefore forces $$ \int_{t_0}^{t_1}\Bigl( \frac{\partial L}{\partial q^i} -\frac{d}{dt}\frac{\partial L}{\partial\dot{q}^i} \Bigr)\eta^i\,dt=0 $$ for every such $\eta$. the fundamental lemma of the calculus of variations implies the Euler-Lagrange equations. coordinate covariance follows because criticality of $S$ is chart-independent: if the integral identity holds in one chart it holds after pushforward by a diffeomorphism of $Q$. $\square$

theorem (energy conservation for autonomous systems). if $L$ does not depend explicitly on $t$, then along every $C^2$ solution of the Euler-Lagrange equations the energy $$ E(q,\dot{q}):=\frac{\partial L}{\partial\dot{q}^i}\dot{q}^i-L $$ is constant.

proof. differentiate $E$ along a solution: $$ \frac{dE}{dt} =\frac{d}{dt}\Bigl(\frac{\partial L}{\partial\dot{q}^i}\Bigr)\dot{q}^i +\frac{\partial L}{\partial\dot{q}^i}\ddot{q}^i -\frac{\partial L}{\partial q^i}\dot{q}^i -\frac{\partial L}{\partial\dot{q}^i}\ddot{q}^i -\frac{\partial L}{\partial t}. $$ the second and fourth terms cancel. Euler-Lagrange replaces $\frac{d}{dt}(\partial L/\partial\dot{q}^i)$ by $\partial L/\partial q^i$, so the first and third terms cancel. if $\partial L/\partial t=0$ one obtains $dE/dt=0$. $\square$

noether's theorem

If a continuous family of transformations of configuration space leaves the Lagrangian invariant, then the generating vector field produces a first integral. This is the geometric origin of the classical conservation laws of energy, momentum, and angular momentum when the corresponding geometric actions (time shifts are handled separately via autonomy) preserve $L$.

definition (symmetry of a lagrangian). a one-parameter group of diffeomorphisms $\Phi_s:Q\to Q$ is a symmetry of $L:TQ\to\mathbb{R}$ if $$ L\bigl(T_q\Phi_s(v)\bigr)=L(v) \qquad\text{for all }v\in T_qQ,\ q\in Q,\ s\in\mathbb{R}. $$ write $\xi_Q(q)=\frac{d}{ds}\big|_{s=0}\Phi_s(q)$ for its infinitesimal generator.

theorem (noether, configuration symmetry). if $\Phi_s$ is a symmetry of a time-independent Lagrangian $L$, then $$ I(q,\dot{q}) =\frac{\partial L}{\partial\dot{q}^i}(q,\dot{q})\,\xi_Q^i(q) $$ is constant along every solution of the Euler-Lagrange equations.

proof. differentiate the invariance identity in $s$ at $s=0$. writing $q_s=\Phi_s\circ q$ for a base curve, $$ 0=\frac{d}{ds}\Big|_{s=0}L(q_s,\dot q_s) =\frac{\partial L}{\partial q^i}\xi_Q^i +\frac{\partial L}{\partial\dot{q}^i}\frac{d}{dt}\xi_Q^i $$ along any lifted motion (the variation of velocity is $\frac{d}{dt}\xi_Q(q)$ because $\xi_Q$ is fixed on $Q$). rearrange as $$ \frac{d}{dt}\Bigl( \frac{\partial L}{\partial\dot{q}^i}\xi_Q^i \Bigr) =\Biggl( \frac{d}{dt}\frac{\partial L}{\partial\dot{q}^i} -\frac{\partial L}{\partial q^i} \Biggr)\xi_Q^i. $$ along Euler-Lagrange solutions the right-hand side vanishes, so $I$ is constant. $\square$

remark (standard consequences). spatial translation symmetry yields conservation of linear momentum; rotational symmetry yields conservation of angular momentum. time-independence of $L$ is the energy theorem already proved, and can be viewed as Noether for a suitably enlarged time reparametrization story once one formulates mechanics on space-time path spaces.

small oscillations and parametric resonance

Near a stable equilibrium $q_0$ one expands the Lagrangian to quadratic order in the displacement $\xi=q-q_0$. For a natural mechanical system the mass metric and potential Hessian produce a linear system $$ M\ddot{\xi}+K\xi=0, $$ where $M$ and $K$ are the mass and stiffness matrices (assumed symmetric, $M$ positive definite, $K$ positive semi-definite at a nondegenerate stable minimum).

definition (normal frequencies). the numbers $\omega\geq 0$ for which there exists $v\neq 0$ with $Kv=\omega^2 Mv$ are the (circular) frequencies of free small oscillation. they are the square roots of the generalized eigenvalues of the pencil $(K,M)$.

remark (parametric resonance). when the stiffness matrix depends periodically on time, the equation becomes a Mathieu or Hill equation. instability bands can appear even while instantaneous frequencies remain real. the phenomenon is linear and geometric: Floquet theory reads the monodromy of a periodic linear system on the phase plane, not a nonlinear resonance of amplitude alone.

rigid-body dynamics

The configuration space of a free rigid body with a fixed point (or free body reduced by translations) may be identified with the rotation group $SO(3)$. In a body-fixed frame the kinetic energy is a quadratic form on the Lie algebra $\mathfrak{so}(3)\simeq\mathbb{R}^3$, and the Euler-Poincaré equations reduce to Euler's rigid-body equations $$ \mathbf{I}\dot{\boldsymbol{\Omega}}+\boldsymbol{\Omega}\times\mathbf{I}\boldsymbol{\Omega}=0. $$ Poinsot's geometric construction realizes the free motion as the rolling without slipping of the inertia ellipsoid on a fixed plane; the polhode and herpolhode curves encode the body and space trajectories of the angular-velocity vector. Coriolis forces appear naturally when equations are written in a rotating frame.

remark (left invariance). the free rigid body is the standard example of a left- invariant Lagrangian on a Lie group. reduction from $T SO(3)$ to the body angular velocities $\boldsymbol{\Omega}$ is the reason the equations live on a three-dimensional space of $\Omega$ rather than on the six-dimensional tangent bundle of $SO(3)$.

dissipative extensions

Friction linear in velocity is incorporated by Rayleigh's dissipation function $R(\dot{q})$. The modified Euler-Lagrange equations become $$ \frac{d}{dt}\Bigl(\frac{\partial L}{\partial\dot{q}}\Bigr) -\frac{\partial L}{\partial q} =-\frac{\partial R}{\partial\dot{q}}. $$ On a Riemannian manifold the resulting dynamics is a gradient flow of the energy with respect to the metric defined by the kinetic energy and the Rayleigh form. The same skeleton underlies the abstract theory of electric circuits, in which voltages and currents are conjugate variables and resistive elements generate a dissipation metric.

variational calculus and optimal control

Carathéodory's approach constructs a field of extremals whose associated Cartan form $$ \theta_L =\frac{\partial L}{\partial\dot{q}^i}\,dq^i +\Bigl(L-\frac{\partial L}{\partial\dot{q}^i}\dot{q}^i\Bigr)dt $$ is closed on the graph of the field. The Hamilton-Jacobi equation is the condition that the Lagrangian coincides with the total differential of a generating function. The same geometric construction, when the admissible velocities are restricted to a control set, yields Pontryagin's maximum principle: an optimal trajectory maximizes the Pontryagin Hamiltonian over the control variables.

remark. optimal control enlarges the pure variational setting to include non-holonomic velocity constraints and bounded controls, while retaining Cartan forms and extremal fields as the common skeleton. chapter material on control theory will return to this once ordinary differential equations and Lie theory are in place.

comparison of formalisms

The Lagrangian picture is natural for unconstrained variational problems and for systems with configuration-space symmetries. The Hamiltonian picture isolates the symplectic geometry of phase space and is indispensable for perturbation theory and integrability. Optimal control enlarges the variational setting while keeping the same geometric backbone of Cartan forms and extremal fields.

Taken together, these constructions convert the empirical Newtonian theory into a coherent differential-geometric framework in which the equations of motion, the conservation laws, and the variational principles are all manifestations of the same underlying manifold structure.

exercises

exercise 1 (central force, angular momentum). for $m\ddot{\mathbf{q}}=-\nabla V(|\mathbf{q}|)$, show again that $\mathbf{L}=\mathbf{q}\times m\dot{\mathbf{q}}$ is constant, then prove that if $\mathbf{L}\neq 0$ the motion lies in a fixed plane. (no appeal to coordinates beyond the definition of the cross product is required until the final planarity sentence.)

exercise 2 (harmonic oscillator, noether). take $L=\frac12 m(\dot x^2+\dot y^2)-\frac12 k(x^2+y^2)$ on $\mathbb{R}^2$ and the one-parameter group of rotations about the origin. compute the Noether integral $I$ of theorem (noether) and identify it with the $z$-component of angular momentum.

exercise 3 (kepler effective potential). for $V(r)=-k/r$ and fixed $\ell>0$, sketch $V_{\mathrm{eff}}$ and determine for which energies $E$ the radial motion is bounded. recover the range of eccentricities of elliptical Kepler orbits from this one-dimensional picture (you may quote $e=\sqrt{1+2E\ell^2/(mk^2)}$ if needed).

exercise 4 (energy from autonomy). derive $dE/dt=0$ for $E=\partial L/\partial\dot{q}\,\dot{q}-L$ when $L$ is time-independent, using only the Euler-Lagrange equations, without referring to the abstract Noether proof above.

exercise 5 (galilean boost, kinetic structure). a free particle has $L=\frac12 m|\dot{\mathbf{q}}|^2$. under a Galilean boost $\mathbf{q}\mapsto\mathbf{q}+\mathbf{v}t$, show that $L$ changes by a total time derivative and thus leaves the equations of motion invariant. compute the corresponding Noether charge and identify it with the center-of-mass constant of motion.