Lagrangian mechanics on the tangent bundle

Intuition [Beginner]

Classical mechanics asks: given a system of particles or a rigid body, what path does it follow over time? The Lagrangian answer reframes the question. Instead of writing forces and integrating Newton's equations, you write a single number for every state — the Lagrangian $L$ — and the system follows whichever path makes the total of $L$ along it as small as possible (more precisely, a critical value).

The state of a mechanical system has two parts: where it is and how fast it is moving. A pendulum's state is its angle and its angular velocity. A planet's state is its position and its velocity. Together these form a single point in a doubled-up space called the tangent bundle — one copy of the configuration space for positions, another for velocities at each position. The Lagrangian eats such a state and returns a number, usually built from kinetic energy minus potential energy.

The Euler-Lagrange equations are what falls out when you ask "for which paths is the total $L$ extremised?" They reproduce Newton's second law for a particle in a potential, give the geodesic equation on a curved surface, and govern almost every mechanical system you can write down.

Visual [Beginner]

A schematic of a configuration space $Q$ with a path $\gamma$ on it, alongside the tangent bundle $TQ$ where the lifted path $(q(t),\dot{q}(t))$ lives. The action $S$ is the area swept by the Lagrangian along the lifted path; the equation of motion says this area is at a critical value.

The picture to keep in mind: the configuration space holds positions, the tangent bundle holds positions-and-velocities together, and the Lagrangian assigns a single number to each such position-velocity pair.

Worked example [Beginner]

Take a single particle of mass $m=2$ moving along a line in the potential $V(q)=q^2$ (a harmonic oscillator with spring constant $2$). The Lagrangian is kinetic energy minus potential energy:

$$L(q,\dot{q})=\frac{1}{2}m{\dot{q}}^2-V(q)={\dot{q}}^2-q^2.$$

The Euler-Lagrange equation rearranges into Newton's second law. The momentum-like quantity is $p=2\dot{q}$; its rate of change is $2\ddot{q}$. The position-derivative of $L$ is $-2q$. Setting "rate of change of $p$ equals position-derivative of $L$" gives $2\ddot{q}=-2q$, equivalent to $\ddot{q}=-q$. The solutions are $q(t)=A\cos (t)+B\sin (t)$ — the ordinary harmonic motion.

What this tells us: the Lagrangian $L=T-V$ encodes the same physical content as Newton's second law $F=ma$, packaged as one scalar function on the tangent bundle. Pick the right $L$ and the equation of motion appears mechanically.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $Q$ be a smooth manifold of dimension $n$, called the configuration space. Its tangent bundle $TQ$ is the smooth $2n$-dimensional manifold whose points are pairs $(q,v)$ with $q\in Q$ and $v\in T_{q}Q$. In local coordinates $(q^1,\dots ,q^n)$ on $Q$, the induced coordinates on $TQ$ are written $(q^1,\dots ,q^n,{\dot{q}}^1,\dots ,{\dot{q}}^n)$ — the ${\dot{q}}^i$ are the components of $v$ in the basis $\{\partial /\partial q^i\}$.

A Lagrangian is a smooth function $L:TQ\to \mathbb{R}$. The action functional assigns to each smooth path $\gamma :[a,b]\to Q$ the real number

$$S[\gamma ]={\int }_a^{b}L(\gamma (t),\dot{\gamma }(t))dt,$$

where $\dot{\gamma }(t)\in T_{\gamma (t)}Q$ is the velocity of $\gamma$ at time $t$.

A variation of $\gamma$ with fixed endpoints is a smooth one-parameter family ${\gamma }_s:[a,b]\to Q$, $s\in (-ϵ,ϵ)$, with ${\gamma }_0=\gamma$ and ${\gamma }_s(a)=\gamma (a)$, ${\gamma }_s(b)=\gamma (b)$ for all $s$. The path $\gamma$ is a critical point of $S$ if $\frac{d}{ds}{|}_{s=0}S[{\gamma }_s]=0$ for every such variation.

In coordinates, the Euler-Lagrange equations for a critical path are

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

The Lagrangian is regular at $(q,v)\in TQ$ if the fibre Hessian $\det \left(\frac{{\partial }^{2}L}{\partial {\dot{q}}^i\partial {\dot{q}}^j}\right)\ne 0$ at $(q,v)$. Regularity at every point makes the Euler-Lagrange system a second-order ODE on $Q$ (equivalently a vector field on $TQ$). The Lagrangian is hyper-regular if the fibre derivative $\mathbb{F}L:TQ\to T^{\ast }Q$, $(q,v)\mapsto (q,\partial L/\partial v)$, is a global diffeomorphism — the hypothesis required for the Hamiltonian formulation.

Counterexamples to common slips

• The fibre Hessian condition is not redundant. $L=q\dot{q}$ on $\mathbb{R}$ has ${\partial }^{2}L/\partial {\dot{q}}^2=0$; the Euler-Lagrange equation collapses to the algebraic identity $\dot{q}-\dot{q}=0$ rather than determining $\ddot{q}$. Singular Lagrangians appear in gauge theory and require constraint analysis (Dirac-Bergmann).
• Adding a total time derivative changes the action by a boundary term, not the equations of motion. If $L^{\prime }=L+\frac{d}{dt}f(q,t)$, then $S^{\prime }[\gamma ]=S[\gamma ]+f(\gamma (b),b)-f(\gamma (a),a)$. Variations with fixed endpoints leave the boundary term invariant, so $L$ and $L^{\prime }$ have identical critical paths.
• The Euler-Lagrange equation transforms covariantly under change of coordinates on $Q$. This is what makes the formulation manifold-intrinsic: solutions are paths in $Q$, not coordinate expressions, and the equations have the same form in any chart.

Key theorem with proof [Intermediate+]

Theorem (variational characterisation of motion). Let $L:TQ\to \mathbb{R}$ be a smooth Lagrangian and $\gamma :[a,b]\to Q$ a smooth path. Then $\gamma$ is a critical point of the action $S[\gamma ]={\int }_a^{b}L(\gamma ,\dot{\gamma })dt$ among paths with fixed endpoints if and only if $\gamma$ satisfies the Euler-Lagrange equations

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

Proof. Work in a coordinate chart on $Q$ in which the path $\gamma$ stays. Let ${\gamma }_s(t)=\gamma (t)+s\eta (t)+O(s^2)$ be a smooth one-parameter variation with $\eta (a)=\eta (b)=0$ (the variation $\eta$ is a vector field along $\gamma$ vanishing at the endpoints). Differentiate $S$ in $s$ at $s=0$:

$$\frac{d}{ds}{|}_{s=0}S[{\gamma }_s]={\int }_a^b\left(\frac{\partial L}{\partial q^i}{\eta }^i+\frac{\partial L}{\partial {\dot{q}}^i}{\dot{\eta }}^i\right)dt.$$

Integrate the second term by parts. The boundary contribution $[\frac{\partial L}{\partial {\dot{q}}^i}{\eta }^i{]}_a^b$ vanishes because $\eta$ vanishes at the endpoints. The bulk gives

$$\frac{d}{ds}{|}_{s=0}S[{\gamma }_s]={\int }_a^b\left(\frac{\partial L}{\partial q^i}-\frac{d}{dt}\frac{\partial L}{\partial {\dot{q}}^i}\right){\eta }^{i}dt.$$

This integral vanishes for every smooth $\eta$ vanishing at the endpoints if and only if the bracketed coefficient vanishes for every $t\in (a,b)$ and every $i$. (The implication "vanishes for every $\eta$ implies coefficient vanishes" is the fundamental lemma of the calculus of variations: if a continuous function $f$ on $[a,b]$ satisfies ${\int }_a^{b}f\eta dt=0$ for every smooth bump function $\eta$ supported in the interior, then $f\equiv 0$ — supported by choosing $\eta$ to be a bump concentrated where $f$ has a definite sign.)

The vanishing of the coefficient is exactly the Euler-Lagrange equation. Conversely, if the Euler-Lagrange equation holds along $\gamma$, the integrand vanishes pointwise and $\gamma$ is a critical point. $\square$

Bridge. The Euler-Lagrange computation here builds toward the coordinate-free symplectic formulation of mechanics: the Poincaré-Cartan one-form ${\theta }_L=\frac{\partial L}{\partial {\dot{q}}^i}d{q}^i$ on $TQ$ encodes the same data, and the Euler-Lagrange equations reappear as ${\iota }_{X}d{\theta }_L=-d{E}_L$ for the energy $E_L={\dot{q}}^i\frac{\partial L}{\partial {\dot{q}}^i}-L$. This identifies Lagrangian dynamics on $TQ$ with the geometry of a particular two-form on $TQ$, the same template that appears again in the Hamiltonian setting 05.01.02 where $T^{\ast }Q$ carries its canonical symplectic form. The Legendre transform $\mathbb{F}L:TQ\to T^{\ast }Q$ identifies primitives with primitives — pulling ${\theta }_{\mathrm{can}}$ on $T^{\ast }Q$ back to ${\theta }_L$ on $TQ$ — and putting these together one sees that classical mechanics is the foundational source of symplectic geometry: every standard Hamiltonian system on $T^{\ast }Q$ is dual to a hyper-regular Lagrangian system on $TQ$.

Exercises [Intermediate+]

Advanced results [Master]

The coordinate-free Lagrangian formalism on $TQ$ admits a structure that mirrors the symplectic formalism on $T^{\ast }Q$ and provides the bridge between them.

The Poincaré-Cartan one-form and the Lagrangian two-form. Given a Lagrangian $L:TQ\to \mathbb{R}$, the Poincaré-Cartan one-form is

$${\theta }_L=\frac{\partial L}{\partial {\dot{q}}^i}d{q}^i\in {\Omega }^1(TQ).$$

Its construction is intrinsic: ${\theta }_L=\mathbb{F}{L}^{\ast }{\theta }_{\mathrm{can}}$ where ${\theta }_{\mathrm{can}}=p_{i}d{q}^i$ is the canonical one-form on $T^{\ast }Q$ and $\mathbb{F}L:TQ\to T^{\ast }Q$, $(q,v)\mapsto (q,\partial L/\partial v)$, is the fibre derivative (Legendre transform). The Lagrangian two-form ${\omega }_L=-d{\theta }_L$ is closed by construction; it is non-degenerate at $(q,v)$ exactly when the fibre Hessian ${\partial }^{2}L/\partial {\dot{q}}^i\partial {\dot{q}}^j$ is non-degenerate at $(q,v)$ — that is, exactly when $L$ is regular at $(q,v)$.

The energy function and the Lagrangian vector field. The energy associated to $L$ is

$$E_L=\Delta L-L={\dot{q}}^i\frac{\partial L}{\partial {\dot{q}}^i}-L,$$

where $\Delta ={\dot{q}}^i\partial /\partial {\dot{q}}^i$ is the Liouville (dilation) vector field on $TQ$ that scales fibres of $TQ\to Q$. For $L=T-V$ with $T$ quadratic in $\dot{q}$, Euler's theorem on homogeneous functions gives $\Delta T=2T$, so $E_L=2T-(T-V)=T+V$ — the total mechanical energy.

Theorem (intrinsic Euler-Lagrange equation). For a regular Lagrangian $L$, the Euler-Lagrange equations on $TQ$ are equivalent to the equation

$${\iota }_{X_L}{\omega }_L=d{E}_L$$

for the unique second-order vector field $X_L$ on $TQ$. The integral curves of $X_L$ project to solutions of the Euler-Lagrange system on $Q$.

This is the cleanest statement: a regular Lagrangian determines a closed two-form ${\omega }_L$ and a function $E_L$, and the equation of motion is the Hamiltonian-style equation for $E_L$ relative to ${\omega }_L$. When $L$ is hyper-regular and $H\circ \mathbb{F}L=E_L$ defines the Hamiltonian on $T^{\ast }Q$, the fibre derivative pulls the canonical symplectic form ${\omega }_{\mathrm{can}}=-d{\theta }_{\mathrm{can}}$ on $T^{\ast }Q$ back to ${\omega }_L$ on $TQ$, and the Lagrangian vector field $X_L$ is $\mathbb{F}L$-related to the Hamiltonian vector field $X_H$.

Regularity, hyper-regularity, and the Legendre map. The fibre derivative $\mathbb{F}L:TQ\to T^{\ast }Q$ is a fibrewise map covering the identity on $Q$. The differential of $\mathbb{F}L$ along the fibre at $(q,v)$ is the Hessian ${\partial }^{2}L/\partial {\dot{q}}^i\partial {\dot{q}}^j$. Regular $L$ means $\mathbb{F}L$ is a local diffeomorphism; hyper-regular means a global diffeomorphism. Hyper-regularity is the standard hypothesis under which Lagrangian and Hamiltonian formulations are interconvertible. The harmonic oscillator and any geodesic Lagrangian are hyper-regular; the relativistic point-particle Lagrangian $L=-m{c}^2\sqrt{1-|v{|}^2/c^2}$ is regular but not hyper-regular without further care; gauge Lagrangians like $L=q\dot{q}$ on $\mathbb{R}$ are not even regular, requiring constraint analysis.

Examples.

• Free particle on ${\mathbb{R}}^n$. $L=\frac{1}{2}m\Vert v{\Vert }^2$. Hyper-regular. $\mathbb{F}L(q,v)=(q,mv)$. $E_L=\frac{1}{2}m\Vert v{\Vert }^2$. Euler-Lagrange: $m\ddot{q}=0$. Solutions are straight lines at constant velocity.

• Particle in a potential. $L=\frac{1}{2}m\Vert v{\Vert }^2-V(q)$. Hyper-regular. $E_L=\frac{1}{2}m\Vert v{\Vert }^2+V(q)$. Euler-Lagrange: $m\ddot{q}=-\nabla V(q)$ — Newton's second law.

• Geodesics on a Riemannian manifold $(Q,g)$. $L=\frac{1}{2}{g}_{ij}(q){\dot{q}}^i{\dot{q}}^j$. Hyper-regular (the fibre Hessian is the metric tensor itself). The fibre derivative is $(q,v)\mapsto (q,g(v,\cdot ))$, the musical isomorphism. Euler-Lagrange yields the geodesic equation ${\ddot{q}}^k+{\Gamma }_{ij}^k{\dot{q}}^i{\dot{q}}^j=0$. The geodesic flow on $TQ$ is the Lagrangian vector field $X_L$.

• Pendulum. $Q=S^1$, $L=\frac{1}{2}m{\ell }^2{\dot{\theta }}^2+mg\ell \cos \theta$. Hyper-regular. $E_L=\frac{1}{2}m{\ell }^2{\dot{\theta }}^2-mg\ell \cos \theta$. Euler-Lagrange: $\ell \ddot{\theta }=-g\sin \theta$. The phase portrait on $T{S}^1$ has the libration-rotation separatrix at energy $E_L=mg\ell$.

Synthesis. The Lagrangian formulation generalises Newton's laws from the metric setting on ${\mathbb{R}}^3$ to mechanics on an arbitrary smooth manifold. The action $S[\gamma ]=\int Ldt$ is the bridge between the path-space view (motion as a critical point of an integral) and the field-on-$TQ$ view (motion as the integral curve of a vector field). This is exactly the relation that recurs throughout symplectic geometry: a closed two-form plus a function determines a vector field, and integrating that vector field reproduces the dynamics. Putting these together one sees that the Lagrangian and Hamiltonian sides of classical mechanics are dual presentations of the same data: the Poincaré-Cartan one-form ${\theta }_L$ on $TQ$ and the canonical one-form ${\theta }_{\mathrm{can}}$ on $T^{\ast }Q$ are pulled to one another by the Legendre map, and the foundational reason classical mechanics gives rise to symplectic geometry is exactly this — the variational principle on $TQ$ identifies critical paths with integral curves of the Lagrangian vector field, and the Legendre map identifies that vector field with the Hamiltonian flow of $E_L\circ \mathbb{F}{L}^{-1}$ on $T^{\ast }Q$. The bridge is constructed once and runs in both directions.

Full proof set [Master]

Lemma (fundamental lemma of the calculus of variations). Let $f:[a,b]\to \mathbb{R}$ be continuous and suppose ${\int }_a^{b}f(t)\eta (t)dt=0$ for every smooth $\eta$ with $\mathrm{supp}(\eta )\subset (a,b)$. Then $f\equiv 0$.

Proof. Suppose $f(t_0)\ne 0$ for some $t_0\in (a,b)$, say $f(t_0)>0$. By continuity, $f>0$ on a neighbourhood $(t_0-\delta ,t_0+\delta )\subset (a,b)$. Choose a smooth bump $\eta \ge 0$ supported in this neighbourhood with $\eta (t_0)>0$. Then ${\int }_a^{b}f\eta dt>0$, contradicting the hypothesis. $\square$

Lemma (variational derivative of the action). Let ${\gamma }_s$ be a one-parameter family of paths $[a,b]\to Q$ with ${\gamma }_0=\gamma$ and variation $\eta =\frac{\partial {\gamma }_s}{\partial s}{|}_{s=0}$. Then

$$\frac{d}{ds}{|}_{s=0}S[{\gamma }_s]={\int }_a^b\left(\frac{\partial L}{\partial q^i}-\frac{d}{dt}\frac{\partial L}{\partial {\dot{q}}^i}\right){\eta }^{i}dt+{\left[\frac{\partial L}{\partial {\dot{q}}^i}{\eta }^i\right]}_a^b.$$

Proof. Differentiate under the integral, $\frac{d}{ds}{|}_{s=0}L({\gamma }_s,{\dot{\gamma }}_s)=(\partial L/\partial q^i){\eta }^i+(\partial L/\partial {\dot{q}}^i){\dot{\eta }}^i$. Integrate the second term by parts: ${\int }_a^b(\partial L/\partial {\dot{q}}^i){\dot{\eta }}^{i}dt=[(\partial L/\partial {\dot{q}}^i){\eta }^i{]}_a^b-{\int }_a^b\frac{d}{dt}(\partial L/\partial {\dot{q}}^i){\eta }^{i}dt$. Combine. $\square$

Theorem (variational characterisation of motion, restated). A path $\gamma$ is a critical point of $S$ among variations with fixed endpoints if and only if $\gamma$ satisfies the Euler-Lagrange equations.

Proof. Combining the two preceding lemmas: $\gamma$ is critical iff ${\int }_a^b(\partial L/\partial q^i-\frac{d}{dt}\partial L/\partial {\dot{q}}^i){\eta }^{i}dt=0$ for every $\eta$ with $\eta (a)=\eta (b)=0$, iff (by the fundamental lemma applied component-wise) the integrand coefficient vanishes pointwise, iff $\gamma$ satisfies the Euler-Lagrange equations. $\square$

Theorem (intrinsic Euler-Lagrange equation). Let $L:TQ\to \mathbb{R}$ be regular. There exists a unique second-order vector field $X_L$ on $TQ$ satisfying ${\iota }_{X_L}{\omega }_L=d{E}_L$ where ${\omega }_L=-d{\theta }_L$ and $E_L=\Delta L-L$. Its integral curves project to Euler-Lagrange solutions on $Q$.

Proof sketch. In coordinates, ${\theta }_L=(\partial L/\partial {\dot{q}}^i)d{q}^i$. Compute ${\omega }_L=-d{\theta }_L=-\frac{{\partial }^{2}L}{\partial q^j\partial {\dot{q}}^i}d{q}^j\wedge d{q}^i-\frac{{\partial }^{2}L}{\partial {\dot{q}}^j\partial {\dot{q}}^i}d{\dot{q}}^j\wedge d{q}^i$. The second-order condition on $X={\dot{q}}^i{\partial }_{q^i}+a^j{\partial }_{{\dot{q}}^j}$ is fixed; the equation ${\iota }_X{\omega }_L=d{E}_L$ at the $d{q}^k$-component reads $\frac{{\partial }^{2}L}{\partial {\dot{q}}^k\partial {\dot{q}}^j}{a}^j+\frac{{\partial }^{2}L}{\partial {\dot{q}}^k\partial q^i}{\dot{q}}^i=\partial L/\partial q^k$. Regularity of $L$ inverts the fibre Hessian to determine $a^j$ uniquely; this $a^j={\ddot{q}}^j$ along integral curves, recovering the Euler-Lagrange equation. The construction of ${\theta }_L$ from the fibre derivative shows the equation is coordinate-independent. $\square$

Connections [Master]

• Hamilton's principle of least action 05.00.02. The variational characterisation proved here is the formal version of the principle of least action: a path is a motion if and only if its action is critical. The principle is the foundational axiom from which the Euler-Lagrange equations are derived.

• Legendre transform / fibre derivative 05.00.03. The Legendre transform $\mathbb{F}L:TQ\to T^{\ast }Q$, $(q,v)\mapsto (q,\partial L/\partial v)$, is the bridge between the Lagrangian formalism on the tangent bundle developed here and the Hamiltonian formalism on the cotangent bundle. The Hamiltonian is $H\circ \mathbb{F}L=E_L$.

• Noether's theorem 05.00.04. Continuous symmetries of a Lagrangian system give conserved quantities along solutions of the Euler-Lagrange equations. Translation symmetry of $L$ in $q^i$ conserves the conjugate momentum $\partial L/\partial {\dot{q}}^i$; time-translation symmetry conserves the energy $E_L$.

• Symplectic manifold 05.01.02. The two-form ${\omega }_L=-d{\theta }_L$ on $TQ$ is symplectic when $L$ is regular; it is the pullback of the canonical symplectic form on $T^{\ast }Q$ along the Legendre map. Lagrangian mechanics is the foundational source of symplectic structures on cotangent bundles, the prototype symplectic manifolds.

• Cotangent bundle 05.02.05. The Lagrangian formulation on $TQ$ is dual, via the Legendre transform, to the Hamiltonian formulation on $T^{\ast }Q$. The Poincaré-Cartan one-form ${\theta }_L$ on $TQ$ pulls back from the canonical one-form ${\theta }_{\mathrm{can}}=p_{i}d{q}^i$ on $T^{\ast }Q$.

• Variational calculus on manifolds 03.04.08. The Euler-Lagrange machinery developed there is the differential-forms framework underlying the proof of the variational characterisation theorem above; this unit specialises to the mechanics setting where the integrand has the specific form $L:TQ\to \mathbb{R}$.

Historical & philosophical context [Master]

Joseph-Louis Lagrange's Mécanique analytique (1788) ^{[Lagrange 1788]} introduced the analytical formulation of mechanics. Working without the modern apparatus of manifolds and tangent bundles, Lagrange wrote the equations of motion in generalised coordinates $(q^1,\dots ,q^n)$ — coordinates adapted to the constraints of the system rather than Cartesian space — and derived the Euler-Lagrange equations from a variational principle. The book opens with the famous remark that it contains no diagrams, and the entire mechanics of constrained systems is reduced to the manipulation of a single function $L=T-V$. This was the first complete coordinate-invariant treatment of mechanics; the manifold structure was implicit in the freedom to choose generalised coordinates, even though smooth manifolds as a formal notion were a nineteenth-century development.

Felix Klein's Erlangen programme (1872) and the rise of differential geometry in the work of Riemann and his successors provided the language in which Lagrange's formalism could be stated intrinsically. Élie Cartan's Leçons sur les invariants intégraux (1922) ^{[Cartan 1922]} introduced the Poincaré-Cartan one-form and gave the modern coordinate-free expression of the Euler-Lagrange equations as a relation between differential forms on phase space. Jean-Marie Souriau's Structure des systèmes dynamiques (1970) ^{[Souriau 1970]} and Ralph Abraham and Jerrold Marsden's Foundations of Mechanics (1978) ^{[Abraham-Marsden]} codified the fully manifold-theoretic version: configuration space is a smooth manifold $Q$, the Lagrangian is a function $L:TQ\to \mathbb{R}$, the Lagrangian two-form ${\omega }_L$ on $TQ$ is the pullback of the canonical symplectic form on $T^{\ast }Q$ along the fibre derivative, and the equation of motion is the integral curve of the Lagrangian vector field $X_L$.

Vladimir Arnold's Mathematical Methods of Classical Mechanics (Russian original 1974, English 1978) ^[Arnold] became the standard pedagogical exposition. Arnold's framing places the geometric objects first — configuration manifold, tangent bundle, Lagrangian — and treats coordinate calculations as derived. The fibre derivative, the variational principle, and the bridge to Hamiltonian mechanics via the Legendre transform are presented as a single coherent development; Lagrange's coordinate computations are the local form of an intrinsic geometric structure.

The conceptual claim — that mechanics is the geometry of $TQ$ together with a Lagrangian — has propagated far beyond classical mechanics. Field theory replaces $TQ$ with a jet bundle and the Lagrangian with a Lagrangian density; gauge theory replaces $L$ with a connection-dependent functional on the space of connections; string theory studies infinite-dimensional analogues on loop and path spaces.

Bibliography [Master]

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