Lagrangian

This article is about the Lagrangian function in Lagrangian mechanics. For other uses, see Lagrangian (disambiguation).

The Lagrangian, L, of a dynamical system is a mathematical function that summarizes the dynamics of the system. For a simple mechanical system, it is the value given by the kinetic energy of the particle minus the potential energy of the particle but it may be generalized to more complex systems. It is used primarily as a key component in the Euler-Lagrange equations to find the path of a particle according to the principle of least action.

The Lagrangian is named after Italian mathematician and astronomer Joseph Louis Lagrange. The concept of a Lagrangian was introduced in a reformulation of classical mechanics introduced by Lagrange known as Lagrangian mechanics in 1788. This reformulation was needed in order to explore mechanics in alternative systems to cartesian coordinates such as polar, cylindrical and spherical coordinates, and where the coordinates do not necessarily refer to position, for which Newton's formulation of classical mechanics is not convenient.

The Lagrangian has since been used in a method to find the acceleration of a particle in a Newtonian gravitational field and to derive the Einstein field equations. This led to its use in applying electromagnetism to curved spacetime and in describing charged black holes. It also has additional uses in Mathematical formalism to find the functional derivative of an action, and in engineering for the analysis and optimisation of dynamic systems.

Lagrangian mechanics

Definition

In classical mechanics, the natural form of the Lagrangian is defined as the kinetic energy, T, of the system minus its potential energy, V. In symbols,

$L=T-V.$

If the Lagrangian of a system is known, then the equations of motion of the system may be obtained by a direct substitution of the expression for the Lagrangian into the Euler–Lagrange equation. The Lagrangian of a given system is not unique, and two Lagrangians describing the same system can differ by the total derivative with respect to time of some function $f(q,t)$, but solving any equivalent Lagrangians will give the same equations of motion.

The Lagrangian formulation

The importance of the Lagrangian is derived from D'Alembert's principle, which asserts that the virtual work of any reversible variations subject to the constraints of the system is zero:

$\sum _{i}(\mathbf {F} _{i}-m_{i}\mathbf {a} _{i})\cdot \delta \mathbf {r} _{i}=0$

The integration of this principle over time for a conservative force is

${\begin{aligned}\int _{t_{1}}^{t_{2}}\sum _{i}(\mathbf {F} _{i}-m_{i}\mathbf {a} _{i})\cdot \delta \mathbf {r} _{i}\,dt&=\int _{t_{1}}^{t_{2}}\delta \left(-V+{\frac {1}{2}}\sum _{i}m_{i}v_{i}^{2}\right)\,dt-\left[\sum _{i}m_{i}\mathbf {v} _{i}\cdot \delta \mathbf {r} _{i}\right]_{t_{1}}^{t_{2}}\\&=\delta \int _{t_{1}}^{t_{2}}L\,dt-\left[\sum _{i}m_{i}\mathbf {v} _{i}\cdot \delta \mathbf {r} _{i}\right]_{t_{1}}^{t_{2}}\end{aligned}}$

If the endpoints are fixed, then the last term vanishes, and we have arrived at Hamilton's principle:

$\delta \int _{t_{1}}^{t_{2}}L\,dt=0.$

Simple example

The trajectory of a thrown ball is characterized by the sum of the Lagrangian values at each time being a (local) minimum.

The Lagrangian L can be calculated at several instants of time t, and a graph of L against t can be drawn. The area under the curve is the action. Any different path between the initial and final positions leads to a larger action than that chosen by nature. Nature chooses the smallest action – this is the Principle of Least Action.

If Nature has defined the mechanics problem of the thrown ball in so elegant a fashion, might She have defined other problems similarly. So it seems now. Indeed, at the present time it appears that we can describe all the fundamental forces in terms of a Lagrangian. The search for Nature's One Equation, which rules all of the universe, has been largely a search for an adequate Lagrangian.

— Robert Adair, The Great Design: Particles, Fields, and Creation

Using only the principle of least action and the Lagrangian we can deduce the correct trajectory, by trial and error or the calculus of variations.

Importance

The Lagrangian formulation of mechanics is important not just for its broad applications, but also for its role in advancing deep understanding of physics. Although Lagrange only sought to describe classical mechanics, the action principle that is used to derive the Lagrange equation was later recognized to be applicable to quantum mechanics as well.

Physical action and quantum-mechanical phase are related via Planck's constant, and the principle of stationary action can be understood in terms of constructive interference of wave functions.

The same principle, and the Lagrangian formalism, are tied closely to Noether's theorem, which connects physical conserved quantities to continuous symmetries of a physical system.

Lagrangian mechanics and Noether's theorem together yield a natural formalism for first quantization by including commutators between certain terms of the Lagrangian equations of motion for a physical system.

Advantages over other methods

• The formulation is not tied to any one coordinate system – rather, any convenient variables may be used to describe the system; these variables are called "generalized coordinates" q_i and may be any quantitative attributes of the system (for example, strength of the magnetic field at a particular location; angle of a pulley; position of a particle in space; or degree of excitation of a particular eigenmode in a complex system) which are functions of the independent variable(s). This trait makes it easy to incorporate constraints into a theory by defining coordinates that only describe states of the system that satisfy the constraints.

• If the Lagrangian is invariant under a symmetry, then the resulting equations of motion are also invariant under that symmetry. This characteristic is very helpful in showing that theories are consistent with either special relativity or general relativity.

Cyclic coordinates and conservation laws

An important property of the Lagrangian is that conservation laws can easily be read off from it. For example, if the Lagrangian $\scriptstyle L$ does not depend on $\scriptstyle q_{i}$ itself ($\scriptstyle q_{i}$ representing a set of generalized coordinates), then the generalized momentum ($\scriptstyle p_{i}$), given by:

$p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}},$

is a conserved quantity, because of Lagrange's equations:

${\dot {p}}_{i}={\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}={\frac {\partial L}{\partial q_{i}}}=0.$

It doesn't matter if $\scriptstyle L$ depends on the time derivative $\scriptstyle {\dot {q}}_{i}$ of that generalized coordinate, since the Lagrangian independence of the coordinate always makes the above partial derivative zero. This is a special case of Noether's theorem. Such coordinates are called "cyclic" or "ignorable".

For example, the conservation of the generalized momentum,

$p_{2}={\frac {\partial L}{\partial {\dot {q}}_{2}}},$

say, can be directly seen if the Lagrangian of the system is of the form

$L(q_{1},q_{3},q_{4},\dots ;{\dot {q}}_{1},{\dot {q}}_{2},{\dot {q}}_{3},{\dot {q}}_{4},\dots ;t)\,.$

Also, if the time t, does not appear in L, then the Hamiltonian, which is related to the Lagrangian by a Legendre transformation, is conserved. This is the energy conservation unless the potential energy depends on velocity, as in electrodynamics.

Explanation

The Lagrangian in many classical systems is a function of generalized coordinates q_i and their velocities dq_i/dt. These coordinates (and velocities) are, in their turn, parametric functions of time. In the classical view, time is an independent variable and q_i (and dq_i/dt) are dependent variables as is often seen in phase space explanations of systems.

The equations of motion obtained from this functional derivative are the Euler–Lagrange equations of this action. For example, in the classical mechanics of particles, the only independent variable is time, t. So the Euler–Lagrange equations are

${\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\varphi }}_{i}}}={\frac {\partial L}{\partial \varphi _{i}}}\,.$

Dynamical systems whose equations of motion are obtainable by means of an action principle on a suitably chosen Lagrangian are known as Lagrangian dynamical systems. Examples of Lagrangian dynamical systems range from the classical version of the Standard Model, to Newton's equations, to purely mathematical problems such as geodesic equations and Plateau's problem.

Mathematical formalism

Suppose we have an n-dimensional manifold, M, and a target manifold, T. Let $\scriptstyle {\mathcal {C}}$ be the configuration space of smooth functions from M to T.

In classical mechanics, in the Hamiltonian formalism, M is the one-dimensional manifold $\scriptstyle \mathbb {R}$, representing time and the target space is the cotangent bundle of space of generalized positions.

Uses in Engineering

Circa 1963 Lagrangians were a general part of the engineering curriculum, but a quarter of a century later, even with the ascendency of dynamical systems, they were dropped as requirements for some engineering programs, and are generally considered to be the domain of theoretical dynamics. Circa 2003 this changed dramatically, and Lagrangians are not only a required part of many ME and EE graduate-level curricula, but also find applications in finance, economics, and biology, mainly as the basis of the formulation of various path integral schemes to facilitate the solution of parabolic partial differential equations via random walks.

Circa 2013, Lagrangians find their way into hundreds of direct engineering solutions, including robotics, turbulent flow analysis (Lagrangian and Eulerian specification of the flow field), signal processing, microscopic component contact and nanotechnology (superlinear convergent augmented Lagrangians), gyroscopic forcing and dissipation, semi-infinite supercomputing (which also involve Lagrange multipliers in the subfield of semi-infinite programming), chemical engineering (specific heat linear Lagrangian interpolation in reaction planning), civil engineering (dynamic analysis of traffic flows), optics engineering and design (Lagrangian and Hamiltonian optics) aerospace (Lagrangian interpolation), force stepping integrators, and even airbag deployment (coupled Eulerian-Lagrangians as well as SELM—the stochastic Eulerian Lagrangian method).

See also

• Calculus of variations
• Covariant classical field theory
• Einstein–Maxwell–Dirac equations
• Euler–Lagrange equation
• Functional derivative
• Functional integral
• Generalized coordinates
• Hamiltonian mechanics

• Lagrangian and Eulerian coordinates
• Lagrangian mechanics
• Lagrangian point
• Lagrangian system
• Noether's theorem
• Onsager–Machlup function
• Principle of least action
• Scalar field theory

References

• David Tong Classical Dynamics (Cambridge lecture notes)