Lagrangian mechanics

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Joseph-Louis Lagrange (1736–1813)

Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788.

In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms, either the Lagrange equations of the first kind, which treat constraints explicitly as extra equations, often using Lagrange multipliers; or the Lagrange equations of the second kind, which incorporate the constraints directly by judicious choice of generalized coordinates. In each case, a mathematical function called the Lagrangian is a function of the generalized coordinates, their time derivatives, and time, and contains the information about the dynamics of the system.

No new physics are necessarily introduced in applying Lagrangian mechanics compared to Newtonian mechanics. It is, however, more mathematically sophisticated and systematic. Newton's laws can include non-conservative forces like friction; however, they must include constraint forces explicitly and are best suited to Cartesian coordinates. Lagrangian mechanics is ideal for systems with conservative forces and for bypassing constraint forces in any coordinate system. Dissipative and driven forces can be accounted for by splitting the external forces into a sum of potential and non-potential forces, leading to a set of modified Euler–Lagrange (EL) equations. Generalized coordinates can be chosen by convenience, to exploit symmetries in the system or the geometry of the constraints, which may simplify solving for the motion of the system. Lagrangian mechanics also reveals conserved quantities and their symmetries in a direct way, as a special case of Noether's theorem.

Lagrangian 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 in his treatise Mécanique analytique, William Rowan Hamilton later developed Hamilton's principle that can be used to derive the Lagrange equation and was later recognized to be applicable to much of fundamental theoretical physics as well, particularly quantum mechanics and the theory of relativity. It can also be applied to other systems by analogy, for instance to coupled electric circuits with inductances and capacitances.

Lagrangian mechanics is widely used to solve mechanical problems in physics and when Newton's formulation of classical mechanics is not convenient. Lagrangian mechanics applies to the dynamics of particles, while fields are described using a Lagrangian density. Lagrange's equations are also used in optimisation problems of dynamic systems. In mechanics, Lagrange's equations of the second kind are used much more than those of the first kind.

Introduction

Bead constrained to move on a frictionless wire. The wire exerts a reaction force C on the bead to keep it on the wire. The non-constraint force N in this case is gravity. Notice the initial position of the wire can lead to different motions.

 

Simple pendulum. Since the rod is rigid, the position of the bob is constrained according to the equation f(x, y) = 0, the constraint force C is the tension in the rod. Again the non-constraint force N in this case is gravity.

Suppose we have a bead sliding around on a wire, or a swinging simple pendulum, etc. If one tracks each of the massive objects (bead, pendulum bob, etc.) as a particle, calculation of the motion of the particle using Newtonian mechanics would require solving for the time-varying constraint force required to keep the particle in the constrained motion (reaction force exerted by the wire on the bead, or tension in the pendulum rod). For the same problem using Lagrangian mechanics, one looks at the path the particle can take and chooses a convenient set of independent generalized coordinates that completely characterize the possible motion of the particle. This choice eliminates the need for the constraint force to enter into the resultant system of equations. There are fewer equations since one is not directly calculating the influence of the constraint on the particle at a given moment.

For a wide variety of physical systems, if the size and shape of a massive object are negligible, it is a useful simplification to treat it as a point particle. For a system of N point particles with masses m₁, m₂, ..., m_N, each particle has a position vector, denoted r₁, r₂, ..., r_N. Cartesian coordinates are often sufficient, so r₁ = (x₁, y₁, z₁), r₂ = (x₂, y₂, z₂) and so on. In three dimensional space, each position vector requires three coordinates to uniquely define the location of a point, so there are 3N coordinates to uniquely define the configuration of the system. These are all specific points in space to locate the particles, a general point in space is written r = (x, y, z). The velocity of each particle is how fast the particle moves along its path of motion, and is the time derivative of its position, thus

$${\displaystyle \mathbf {v} _{1}={\frac {d\mathbf {r} _{1}}{dt}},\mathbf {v} _{2}={\frac {d\mathbf {r} _{2}}{dt}},\ldots ,\mathbf {v} _{N}={\frac {d\mathbf {r} _{N}}{dt}}}$$

In Newtonian mechanics, the equations of motion are given by Newton's laws. The second law "net force equals mass times acceleration",

$${\displaystyle \sum \mathbf {F} =m{\frac {d^{2}\mathbf {r} }{dt^{2}}}}$$

applies to each particle. For an N particle system in 3 dimensions, there are 3N second order ordinary differential equations in the positions of the particles to solve for.

Instead of forces, Lagrangian mechanics uses the energies in the system. The central quantity of Lagrangian mechanics is the Lagrangian, a function which summarizes the dynamics of the entire system. Overall, the Lagrangian has units of energy, but no single expression for all physical systems. Any function which generates the correct equations of motion, in agreement with physical laws, can be taken as a Lagrangian. It is nevertheless possible to construct general expressions for large classes of applications. The non-relativistic Lagrangian for a system of particles can be defined by

${\displaystyle L=T-V}$

where

${\displaystyle T={\frac {1}{2}}\sum _{k=1}^{N}m_{k}v_{k}^{2}}$

is the total kinetic energy of the system, equalling the sum Σ of the kinetic energies of the particles, and V is the potential energy of the system.

Kinetic energy is the energy of the system's motion, and v_k² = v_k · v_k is the magnitude squared of velocity, equivalent to the dot product of the velocity with itself. The kinetic energy is a function only of the velocities v_k, not the positions r_k nor time t, so T = T(v₁, v₂, ...).

The potential energy of the system reflects the energy of interaction between the particles, i.e. how much energy any one particle will have due to all the others and other external influences. For conservative forces (e.g. Newtonian gravity), it is a function of the position vectors of the particles only, so V = V(r₁, r₂, ...). For those non-conservative forces which can be derived from an appropriate potential (e.g. electromagnetic potential), the velocities will appear also, V = V(r₁, r₂, ..., v₁, v₂, ...). If there is some external field or external driving force changing with time, the potential will change with time, so most generally V = V(r₁, r₂, ..., v₁, v₂, ..., t).

The above form of L does not hold in relativistic Lagrangian mechanics, and must be replaced by a function consistent with special or general relativity. Also, for dissipative forces another function must be introduced alongside L.

One or more of the particles may each be subject to one or more holonomic constraints, such a constraint is described by an equation of the form f(r, t) = 0. If the number of constraints in the system is C, then each constraint has an equation, f₁(r, t) = 0, f₂(r, t) = 0, ... f_C(r, t) = 0, each could apply to any of the particles. If particle k is subject to constraint i, then f_i(r_k, t) = 0. At any instant of time, the coordinates of a constrained particle are linked together and not independent. The constraint equations determine the allowed paths the particles can move along, but not where they are or how fast they go at every instant of time. Nonholonomic constraints depend on the particle velocities, accelerations, or higher derivatives of position. Lagrangian mechanics can only be applied to systems whose constraints, if any, are all holonomic. Three examples of nonholonomic constraints are: when the constraint equations are nonintegrable, when the constraints have inequalities, or with complicated non-conservative forces like friction. Nonholonomic constraints require special treatment, and one may have to revert to Newtonian mechanics, or use other methods.

If T or V or both depend explicitly on time due to time-varying constraints or external influences, the Lagrangian L(r₁, r₂, ... v₁, v₂, ... t) is explicitly time-dependent. If neither the potential nor the kinetic energy depend on time, then the Lagrangian L(r₁, r₂, ... v₁, v₂, ...) is explicitly independent of time. In either case, the Lagrangian will always have implicit time-dependence through the generalized coordinates.

With these definitions Lagrange's equations of the first kind are

Lagrange's equations (First kind) ${\displaystyle {\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0}$

where k = 1, 2, ..., N labels the particles, there is a Lagrange multiplier λ_i for each constraint equation f_i, and

${\displaystyle {\frac {\partial }{\partial \mathbf {r} _{k}}}\equiv \left({\frac {\partial }{\partial x_{k}}},{\frac {\partial }{\partial y_{k}}},{\frac {\partial }{\partial z_{k}}}\right)\,,\quad {\frac {\partial }{\partial {\dot {\mathbf {r} }}_{k}}}\equiv \left({\frac {\partial }{\partial {\dot {x}}_{k}}},{\frac {\partial }{\partial {\dot {y}}_{k}}},{\frac {\partial }{\partial {\dot {z}}_{k}}}\right)}$

are each shorthands for a vector of partial derivatives ∂/∂ with respect to the indicated variables (not a derivative with respect to the entire vector).^{[nb 1]} Each overdot is a shorthand for a time derivative. This procedure does increase the number of equations to solve compared to Newton's laws, from 3N to 3N + C, because there are 3N coupled second order differential equations in the position coordinates and multipliers, plus C constraint equations. However, when solved alongside the position coordinates of the particles, the multipliers can yield information about the constraint forces. The coordinates do not need to be eliminated by solving the constraint equations.

In the Lagrangian, the position coordinates and velocity components are all independent variables, and derivatives of the Lagrangian are taken with respect to these separately according to the usual differentiation rules (e.g. the derivative of L with respect to the z-velocity component of particle 2, v_z2 = dz₂/dt, is just that; no awkward chain rules or total derivatives need to be used to relate the velocity component to the corresponding coordinate z₂).

In each constraint equation, one coordinate is redundant because it is determined from the other two. The number of independent coordinates is therefore n = 3N − C. We can transform each position vector to a common set of n generalized coordinates, conveniently written as an n-tuple q = (q₁, q₂, ... q_n), by expressing each position vector, and hence the position coordinates, as functions of the generalized coordinates and time,

${\displaystyle \mathbf {r} _{k}=\mathbf {r} _{k}(\mathbf {q} ,t)=(x_{k}(\mathbf {q} ,t),y_{k}(\mathbf {q} ,t),z_{k}(\mathbf {q} ,t),t)\,.}$

The vector q is a point in the configuration space of the system. The time derivatives of the generalized coordinates are called the generalized velocities, and for each particle the transformation of its velocity vector, the total derivative of its position with respect to time, is

${\displaystyle {\dot {q}}_{j}={\frac {\mathrm {d} q_{j}}{\mathrm {d} t}}\,,\quad \mathbf {v} _{k}=\sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}{\dot {q}}_{j}+{\frac {\partial \mathbf {r} _{k}}{\partial t}}\,.}$

Given this v_k, the kinetic energy in generalized coordinates depends on the generalized velocities, generalized coordinates, and time if the position vectors depend explicitly on time due to time-varying constraints, so T = T(q, dq/dt, t).

With these definitions we have the Euler–Lagrange equations, or Lagrange's equations of the second kind

Lagrange's equations (Second kind) ${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)={\frac {\partial L}{\partial q_{j}}}}$

are mathematical results from the calculus of variations, which can also be used in mechanics. Substituting in the Lagrangian L(q, dq/dt, t), gives the equations of motion of the system. The number of equations has decreased compared to Newtonian mechanics, from 3N to n = 3N − C coupled second order differential equations in the generalized coordinates. These equations do not include constraint forces at all, only non-constraint forces need to be accounted for.

Although the equations of motion include partial derivatives, the results of the partial derivatives are still ordinary differential equations in the position coordinates of the particles. The total time derivative denoted d/dt often involves implicit differentiation. Both equations are linear in the Lagrangian, but will generally be nonlinear coupled equations in the coordinates.

From Newtonian to Lagrangian mechanics

Newton's laws

Isaac Newton (1642—1727)

For simplicity, Newton's laws can be illustrated for one particle without much loss of generality (for a system of N particles, all of these equations apply to each particle in the system). The equation of motion for particle of mass m is Newton's second law of 1687, in modern vector notation

${\displaystyle \mathbf {F} =m\mathbf {a} \,,}$

where a is its acceleration and F the resultant force acting on it. In three spatial dimensions, this is a system of three coupled second order ordinary differential equations to solve, since there are three components in this vector equation. The solutions are the position vectors r of the particles at time t, subject to the initial conditions of r and v when t = 0.

Newton's laws are easy to use in Cartesian coordinates, but Cartesian coordinates are not always convenient, and for other coordinate systems the equations of motion can become complicated. In a set of curvilinear coordinates ξ = (ξ¹, ξ², ξ³), the law in tensor index notation is the "Lagrangian form"

${\displaystyle F^{a}=m\left({\frac {\mathrm {d} ^{2}\xi ^{a}}{\mathrm {d} t^{2}}}+\Gamma ^{a}{}_{bc}{\frac {\mathrm {d} \xi ^{b}}{\mathrm {d} t}}{\frac {\mathrm {d} \xi ^{c}}{\mathrm {d} t}}\right)=g^{ak}\left({\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {\xi }}^{k}}}-{\frac {\partial T}{\partial \xi ^{k}}}\right)\,,\quad {\dot {\xi }}^{a}\equiv {\frac {\mathrm {d} \xi ^{a}}{\mathrm {d} t}}\,,}$

where F^a is the ath contravariant components of the resultant force acting on the particle, Γ^a_bc are the Christoffel symbols of the second kind,

${\displaystyle T={\frac {1}{2}}mg_{bc}{\frac {\mathrm {d} \xi ^{b}}{\mathrm {d} t}}{\frac {\mathrm {d} \xi ^{c}}{\mathrm {d} t}}\,,}$

is the kinetic energy of the particle, and g_bc the covariant components of the metric tensor of the curvilinear coordinate system. All the indices a, b, c, each take the values 1, 2, 3. Curvilinear coordinates are not the same as generalized coordinates.

It may seem like an overcomplication to cast Newton's law in this form, but there are advantages. The acceleration components in terms of the Christoffel symbols can be avoided by evaluating derivatives of the kinetic energy instead. If there is no resultant force acting on the particle, F = 0, it does not accelerate, but moves with constant velocity in a straight line. Mathematically, the solutions of the differential equation are geodesics, the curves of extremal length between two points in space (These may end up being minimal so the shortest paths, but that is not necessary). In flat 3d real space the geodesics are simply straight lines. So for a free particle, Newton's second law coincides with the geodesic equation, and states free particles follow geodesics, the extremal trajectories it can move along. If the particle is subject to forces, F ≠ 0, the particle accelerates due to forces acting on it, and deviates away from the geodesics it would follow if free. With appropriate extensions of the quantities given here in flat 3d space to 4d curved spacetime, the above form of Newton's law also carries over to Einstein's general relativity, in which case free particles follow geodesics in curved spacetime that are no longer "straight lines" in the ordinary sense.

However, we still need to know the total resultant force F acting on the particle, which in turn requires the resultant non-constraint force N plus the resultant constraint force C,

${\displaystyle \mathbf {F} =\mathbf {C} +\mathbf {N} \,.}$

The constraint forces can be complicated, since they will generally depend on time. Also, if there are constraints, the curvilinear coordinates are not independent but related by one or more constraint equations.

The constraint forces can either be eliminated from the equations of motion so only the non-constraint forces remain, or included by including the constraint equations in the equations of motion.

D'Alembert's principle

Jean d'Alembert (1717—1783)

One degree of freedom.

Two degrees of freedom.

Constraint force C and virtual displacement δr for a particle of mass m confined to a curve. The resultant non-constraint force is N.

A fundamental result in analytical mechanics is D'Alembert's principle, introduced in 1708 by Jacques Bernoulli to understand static equilibrium, and developed by D'Alembert in 1743 to solve dynamical problems. The principle asserts for N particles

${\displaystyle \sum _{k=1}^{N}(\mathbf {N} _{k}+\mathbf {C} _{k}-m_{k}\mathbf {a} _{k})\cdot \delta \mathbf {r} _{k}=0\,.}$

The δr_k are virtual displacements, by definition they are infinitesimal changes in the configuration of the system consistent with the constraint forces acting on the system at an instant of time, i.e. in such a way that the constraint forces maintain the constrained motion. They are not the same as the actual displacements in the system, which are caused by the resultant constraint and non-constraint forces acting on the particle to accelerate and move it. Virtual work is the work done along a virtual displacement for any force (constraint or non-constraint).

Since the constraint forces act perpendicular to the motion of each particle in the system to maintain the constraints, the total virtual work by the constraint forces acting on the system is zero;

${\displaystyle \sum _{k=1}^{N}\mathbf {C} _{k}\cdot \delta \mathbf {r} _{k}=0\,,}$

so that

${\displaystyle \sum _{k=1}^{N}(\mathbf {N} _{k}-m_{k}\mathbf {a} _{k})\cdot \delta \mathbf {r} _{k}=0\,.}$

Thus D'Alembert's principle allows us to concentrate on only the applied non-constraint forces, and exclude the constraint forces in the equations of motion. The form shown is also independent of the choice of coordinates. However, it cannot be readily used to set up the equations of motion in an arbitrary coordinate system since the displacements δr_k might be connected by a constraint equation, which prevents us from setting the N individual summands to 0. We will therefore seek a system of mutually independent coordinates for which the total sum will be 0 if and only if the individual summands are 0. Setting each of the summands to 0 will eventually give us our separated equations of motion.

Equations of motion from D'Alembert's principle

If there are constraints on particle k, then since the coordinates of the position r_k = (x_k, y_k, z_k) are linked together by a constraint equation, so are those of the virtual displacements δr_k = (δx_k, δy_k, δz_k). Since the generalized coordinates are independent, we can avoid the complications with the δr_k by converting to virtual displacements in the generalized coordinates. These are related in the same form as a total differential,

${\displaystyle \delta \mathbf {r} _{k}=\sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\delta q_{j}\,.}$

There is no partial time derivative with respect to time multiplied by a time increment, since this is a virtual displacement, one along the constraints in an instant of time.

The first term in D'Alembert's principle above is the virtual work done by the non-constraint forces N_k along the virtual displacements δr_k, and can without loss of generality be converted into the generalized analogues by the definition of generalized forces

${\displaystyle Q_{j}=\sum _{k=1}^{N}\mathbf {N} _{k}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\,,}$

so that

${\displaystyle \sum _{k=1}^{N}\mathbf {N} _{k}\cdot \delta \mathbf {r} _{k}=\sum _{k=1}^{N}\mathbf {N} _{k}\cdot \sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\delta q_{j}=\sum _{j=1}^{n}Q_{j}\delta q_{j}\,.}$

This is half of the conversion to generalized coordinates. It remains to convert the acceleration term into generalized coordinates, which is not immediately obvious. Recalling the Lagrange form of Newton's second law, the partial derivatives of the kinetic energy with respect to the generalized coordinates and velocities can be found to give the desired result;

${\displaystyle \sum _{k=1}^{N}m_{k}\mathbf {a} _{k}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}\,.}$

Now D'Alembert's principle is in the generalized coordinates as required,

${\displaystyle \sum _{j=1}^{n}\left[Q_{j}-\left({\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}\right)\right]\delta q_{j}=0\,,}$

and since these virtual displacements δq_j are independent and nonzero, the coefficients can be equated to zero, resulting in Lagrange's equations or the generalized equations of motion,

${\displaystyle Q_{j}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}}$

These equations are equivalent to Newton's laws for the non-constraint forces. The generalized forces in this equation are derived from the non-constraint forces only – the constraint forces have been excluded from D'Alembert's principle and do not need to be found. The generalized forces may be non-conservative, provided they satisfy D'Alembert's principle.

Euler–Lagrange equations and Hamilton's principle

As the system evolves, q traces a path through configuration space (only some are shown). The path taken by the system (red) has a stationary action (δS = 0) under small changes in the configuration of the system (δq).

For a non-conservative force which depends on velocity, it may be possible to find a potential energy function V that depends on positions and velocities. If the generalized forces Q_i can be derived from a potential V such that

${\displaystyle Q_{j}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial V}{\partial {\dot {q}}_{j}}}-{\frac {\partial V}{\partial q_{j}}}\,,}$

equating to Lagrange's equations and defining the Lagrangian as L = T − V obtains Lagrange's equations of the second kind or the Euler–Lagrange equations of motion

${\displaystyle {\frac {\partial L}{\partial q_{j}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}=0\,.}$

However, the Euler–Lagrange equations can only account for non-conservative forces if a potential can be found as shown. This may not always be possible for non-conservative forces, and Lagrange's equations do not involve any potential, only generalized forces; therefore they are more general than the Euler–Lagrange equations.

The Euler–Lagrange equations also follow from the calculus of variations. The variation of the Lagrangian is

${\displaystyle \delta L=\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}\delta q_{j}+{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta {\dot {q}}_{j}\right)\,,\quad \delta {\dot {q}}_{j}\equiv \delta {\frac {\mathrm {d} q_{j}}{\mathrm {d} t}}\equiv {\frac {\mathrm {d} (\delta q_{j})}{\mathrm {d} t}}\,,}$

which has a similar form to the total differential of L, but the virtual displacements and their time derivatives replace differentials, and there is no time increment in accordance with the definition of the virtual displacements. An integration by parts with respect to time can transfer the time derivative of δq_j to the ∂L/∂(dq_j/dt), in the process exchanging d(δq_j)/dt for δq_j, allowing the independent virtual displacements to be factorized from the derivatives of the Lagrangian,

${\displaystyle \int _{t_{1}}^{t_{2}}\delta L\,\mathrm {d} t=\sum _{j=1}^{n}\left[{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta q_{j}\right]_{t_{1}}^{t_{2}}+\int _{t_{1}}^{t_{2}}\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)\delta q_{j}\,\mathrm {d} t\,.}$

Now, if the condition δq_j(t₁) = δq_j(t₂) = 0 holds for all j, the terms not integrated are zero. If in addition the entire time integral of δL is zero, then because the δq_j are independent, and the only way for a definite integral to be zero is if the integrand equals zero, each of the coefficients of δq_j must also be zero. Then we obtain the equations of motion. This can be summarized by Hamilton's principle;

${\displaystyle \int _{t_{1}}^{t_{2}}\delta L\,\mathrm {d} t=0\,.}$

The time integral of the Lagrangian is another quantity called the action, defined as

${\displaystyle S=\int _{t_{1}}^{t_{2}}L\,\mathrm {d} t\,,}$

which is a functional; it takes in the Lagrangian function for all times between t₁ and t₂ and returns a scalar value. Its dimensions are the same as [ angular momentum ], [energy]·[time], or [length]·[momentum]. With this definition Hamilton's principle is

${\displaystyle \delta S=0\,.}$

Thus, instead of thinking about particles accelerating in response to applied forces, one might think of them picking out the path with a stationary action, with the end points of the path in configuration space held fixed at the initial and final times. Hamilton's principle is sometimes referred to as the principle of least action, however the action functional need only be stationary, not necessarily a maximum or a minimum value. Any variation of the functional gives an increase in the functional integral of the action.

Historically, the idea of finding the shortest path a particle can follow subject to a force motivated the first applications of the calculus of variations to mechanical problems, such as the Brachistochrone problem solved by Jean Bernoulli in 1696, as well as Leibniz, Daniel Bernoulli, L'Hôpital around the same time, and Newton the following year. Newton himself was thinking along the lines of the variational calculus, but did not publish. These ideas in turn lead to the variational principles of mechanics, of Fermat, Maupertuis, Euler, Hamilton, and others.

Hamilton's principle can be applied to nonholonomic constraints if the constraint equations can be put into a certain form, a linear combination of first order differentials in the coordinates. The resulting constraint equation can be rearranged into first order differential equation. This will not be given here.

Lagrange multipliers and constraints

The Lagrangian L can be varied in the Cartesian r_k coordinates, for N particles,

${\displaystyle \int _{t_{1}}^{t_{2}}\sum _{k=1}^{N}\left({\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}\right)\cdot \delta \mathbf {r} _{k}\,\mathrm {d} t=0\,.}$

Hamilton's principle is still valid even if the coordinates L is expressed in are not independent, here r_k, but the constraints are still assumed to be holonomic. As always the end points are fixed δr_k(t₁) = δr_k(t₂) = 0 for all k. What cannot be done is to simply equate the coefficients of δr_k to zero because the δr_k are not independent. Instead, the method of Lagrange multipliers can be used to include the constraints. Multiplying each constraint equation f_i(r_k, t) = 0 by a Lagrange multiplier λ_i for i = 1, 2, ..., C, and adding the results to the original Lagrangian, gives the new Lagrangian

${\displaystyle L'=L(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,{\dot {\mathbf {r} }}_{1},{\dot {\mathbf {r} }}_{2},\ldots ,t)+\sum _{i=1}^{C}\lambda _{i}(t)f_{i}(\mathbf {r} _{k},t)\,.}$

The Lagrange multipliers are arbitrary functions of time t, but not functions of the coordinates r_k, so the multipliers are on equal footing with the position coordinates. Varying this new Lagrangian and integrating with respect to time gives

${\displaystyle \int _{t_{1}}^{t_{2}}\delta L'\mathrm {d} t=\int _{t_{1}}^{t_{2}}\sum _{k=1}^{N}\left({\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}\right)\cdot \delta \mathbf {r} _{k}\,\mathrm {d} t=0\,.}$

The introduced multipliers can be found so that the coefficients of δr_k are zero, even though the r_k are not independent. The equations of motion follow. From the preceding analysis, obtaining the solution to this integral is equivalent to the statement

${\displaystyle {\frac {\partial L'}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {\mathbf {r} }}_{k}}}=0\quad \Rightarrow \quad {\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0\,,}$

which are Lagrange's equations of the first kind. Also, the λ_i Euler-Lagrange equations for the new Lagrangian return the constraint equations

${\displaystyle {\frac {\partial L'}{\partial \lambda _{i}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {\lambda }}_{i}}}=0\quad \Rightarrow \quad f_{i}(\mathbf {r} _{k},t)=0\,.}$

For the case of a conservative force given by the gradient of some potential energy V, a function of the r_k coordinates only, substituting the Lagrangian L = T − V gives

${\displaystyle \underbrace {{\frac {\partial T}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {\mathbf {r} }}_{k}}}} _{-\mathbf {F} _{k}}+\underbrace {-{\frac {\partial V}{\partial \mathbf {r} _{k}}}} _{\mathbf {N} _{k}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0\,,}$

and identifying the derivatives of kinetic energy as the (negative of the) resultant force, and the derivatives of the potential equaling the non-constraint force, it follows the constraint forces are

${\displaystyle \mathbf {C} _{k}=\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}\,,}$

thus giving the constraint forces explicitly in terms of the constraint equations and the Lagrange multipliers.

Properties of the Euler–Lagrange equation

In some cases, the Lagrangian has properties which can provide information about the system without solving the equations of motion. These follow from Lagrange's equations of the second kind.

Non-uniqueness

The Lagrangian of a given system is not unique. A Lagrangian L can be multiplied by a nonzero constant a, an arbitrary constant b can be added, and the new Lagrangian aL + b will describe exactly the same motion as L. A less obvious result is that two Lagrangians describing the same system can differ by the total derivative (not partial) of some function f(q, t) with respect to time;

${\displaystyle L'=L+{\frac {\mathrm {d} f(\mathbf {q} ,t)}{\mathrm {d} t}}\,.}$

Each Lagrangian will obtain exactly the same equations of motion.

Invariance under point transformations

Given a set of generalized coordinates q, if we change these variables to a new set of generalized coordinates s according to a point transformation q = q(s, t), the new Lagrangian L′ is a function of the new coordinates

${\displaystyle L(\mathbf {q} (\mathbf {s} ,t),{\dot {\mathbf {q} }}(\mathbf {s} ,{\dot {\mathbf {s} }},t),t)=L'(\mathbf {s} ,{\dot {\mathbf {s} }},t)\,,}$

and by the chain rule for partial differentiation, Lagrange's equations are invariant under this transformation;

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {s}}_{i}}}={\frac {\partial L'}{\partial s_{i}}}\,.}$

This may simplify the equations of motion.

Cyclic coordinates and conserved momenta

An important property of the Lagrangian is that conserved quantities can easily be read off from it. The generalized momentum "canonically conjugate to" the coordinate q_i is defined by

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

If the Lagrangian L does not depend on some coordinate q_i, it follows immediately from the Euler–Lagrange equations that

${\displaystyle {\dot {p}}_{i}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}={\frac {\partial L}{\partial q_{i}}}=0\,.}$

and integrating shows the corresponding generalized momentum equals a constant, a conserved quantity. This is a special case of Noether's theorem. Such coordinates are called "cyclic" or "ignorable".

For example, a system may have a Lagrangian

${\displaystyle L(r,\theta ,{\dot {s}},{\dot {z}},{\dot {r}},{\dot {\theta }},{\dot {\phi }},t)\,,}$

where r and z are lengths along straight lines, s is an arc length along some curve, and θ and φ are angles. Notice z, s, and φ are all absent in the Lagrangian even though their velocities are not. Then the momenta

${\displaystyle p_{z}={\frac {\partial L}{\partial {\dot {z}}}}\,,\quad p_{s}={\frac {\partial L}{\partial {\dot {s}}}}\,,\quad p_{\phi }={\frac {\partial L}{\partial {\dot {\phi }}}}\,,}$

are all conserved quantities. The units and nature of each generalized momentum will depend on the corresponding coordinate; in this case p_z is a translational momentum in the z direction, p_s is also a translational momentum along the curve s is measured, and p_φ is an angular momentum in the plane the angle φ is measured in. However complicated the motion of the system is, all the coordinates and velocities will vary in such a way that these momenta are conserved.

Energy conservation

Taking the total derivative of the Lagrangian L = T − V with respect to time leads to the general result

${\displaystyle -{\frac {\partial L}{\partial t}}={\frac {\mathrm {d} }{\mathrm {d} t}}\left(\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-L\right)\,.}$

If the entire Lagrangian is explicitly independent of time, it follows the partial time derivative of the Lagrangian is zero, ∂L/∂t = 0, so the quantity under the total time derivative in brackets

${\displaystyle E=\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-L}$

must be a constant for all times during the motion of the system, and it also follows the kinetic energy is a homogenous function of degree 2 in the generalized velocities. If in addition the potential V is only a function of coordinates and independent of velocities, it follows by direct calculation, or use of Euler's theorem for homogenous functions, that

${\displaystyle \sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}=\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial T}{\partial {\dot {q}}_{i}}}=2T\,.}$

Under all these circumstances, the constant

${\displaystyle E=T+V}$

is the total conserved energy of the system. The kinetic and potential energies still change as the system evolves, but the motion of the system will be such that their sum, the total energy, is constant. This is a valuable simplification, since the energy E is a constant of integration that counts as an arbitrary constant for the problem, and it may be possible to integrate the velocities from this energy relation to solve for the coordinates. In the case the velocity or kinetic energy or both depends on time, then the energy is not conserved.

Mechanical similarity

If the potential energy is a homogeneous function of the coordinates and independent of time, and all position vectors are scaled by the same nonzero constant α, r_k′ = αr_k, so that

${\displaystyle V(\alpha \mathbf {r} _{1},\alpha \mathbf {r} _{2},\ldots ,\alpha \mathbf {r} _{N})=\alpha ^{N}V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N})}$

and time is scaled by a factor β, t′ = βt, then the velocities v_k are scaled by a factor of α/β and the kinetic energy T by (α/β)². The entire Lagrangian has been scaled by the same factor if

${\displaystyle {\frac {\alpha ^{2}}{\beta ^{2}}}=\alpha ^{N}\quad \Rightarrow \quad \beta =\alpha ^{1-N/2}\,.}$

Since the lengths and times have been scaled, the trajectories of the particles in the system follow geometrically similar paths differing in size. The length l traversed in time t in the original trajectory corresponds to a new length l′ traversed in time t′ in the new trajectory, given by the ratios

${\displaystyle {\frac {t'}{t}}=\left({\frac {l'}{l}}\right)^{1-N/2}\,.}$

Interacting particles

For a given system, if two subsystems A and B are non-interacting, the Lagrangian L of the overall system is the sum of the Lagrangians L_A and L_B for the subsystems:

${\displaystyle L=L_{A}+L_{B}\,.}$

If they do interact this is not possible. In some situations, it may be possible to separate the Lagrangian of the system L into the sum of non-interacting Lagrangians, plus another Lagrangian L_AB containing information about the interaction,

${\displaystyle L=L_{A}+L_{B}+L_{AB}\,.}$

This may be physically motivated by taking the non-interacting Lagrangians to be kinetic energies only, while the interaction Lagrangian is the system's total potential energy. Also, in the limiting case of negligible interaction, L_AB tends to zero reducing to the non-interacting case above.

The extension to more than two non-interacting subsystems is straightforward – the overall Lagrangian is the sum of the separate Lagrangians for each subsystem. If there are interactions, then interaction Lagrangians may be added.

Examples

The following examples apply Lagrange's equations of the second kind to mechanical problems.

Conservative force

A particle of mass m moves under the influence of a conservative force derived from the gradient ∇ of a scalar potential,

${\displaystyle \mathbf {F} =-\nabla V(\mathbf {r} )\,.}$

If there are more particles, in accordance with the above results, the total kinetic energy is a sum over all the particle kinetic energies, and the potential is a function of all the coordinates.

Cartesian coordinates

The Lagrangian of the particle can be written

${\displaystyle L(x,y,z,{\dot {x}},{\dot {y}},{\dot {z}})={\frac {1}{2}}m({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})-V(x,y,z)\,.}$

The equations of motion for the particle are found by applying the Euler–Lagrange equation, for the x coordinate

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {x}}}}\right)={\frac {\partial L}{\partial x}}\,,}$

with derivatives

${\displaystyle {\frac {\partial L}{\partial x}}=-{\frac {\partial V}{\partial x}}\,,\quad {\frac {\partial L}{\partial {\dot {x}}}}=m{\dot {x}}\,,\quad {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {x}}}}\right)=m{\ddot {x}}\,,}$

hence

${\displaystyle m{\ddot {x}}=-{\frac {\partial V}{\partial x}}\,,}$

and similarly for the y and z coordinates. Collecting the equations in vector form we find

${\displaystyle m{\ddot {\mathbf {r} }}=-\nabla V}$

which is Newton's second law of motion for a particle subject to a conservative force.

Polar coordinates in 2d and 3d

The Lagrangian for the above problem in spherical coordinates, with a central potential, is

${\displaystyle L={\frac {m}{2}}({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}+r^{2}\sin ^{2}\theta \,{\dot {\varphi }}^{2})-V(r)\,,}$

so the Euler–Lagrange equations are

${\displaystyle m{\ddot {r}}-mr({\dot {\theta }}^{2}+\sin ^{2}\theta \,{\dot {\varphi }}^{2})+{\frac {\partial V}{\partial r}}=0\,,}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(mr^{2}{\dot {\theta }})-mr^{2}\sin \theta \cos \theta \,{\dot {\varphi }}^{2}=0\,,}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(mr^{2}\sin ^{2}\theta \,{\dot {\varphi }})=0\,.}$

The φ coordinate is cyclic since it does not appear in the Lagrangian, so the conserved momentum in the system is the angular momentum

${\displaystyle p_{\varphi }={\frac {\partial L}{\partial {\dot {\varphi }}}}=mr^{2}\sin ^{2}\theta {\dot {\varphi }}\,,}$

in which r, θ and dφ/dt can all vary with time, but only in such a way that p_φ is constant.

Pendulum on a movable support

Sketch of the situation with definition of the coordinates (click to enlarge)

Consider a pendulum of mass m and length ℓ, which is attached to a support with mass M, which can move along a line in the x-direction. Let x be the coordinate along the line of the support, and let us denote the position of the pendulum by the angle θ from the vertical. The coordinates and velocity components of the pendulum bob are

${\displaystyle {\begin{array}{rll}&x_{\mathrm {pend} }=x+\ell \sin \theta &\quad \Rightarrow \quad {\dot {x}}_{\mathrm {pend} }={\dot {x}}+\ell {\dot {\theta }}\cos \theta \\&y_{\mathrm {pend} }=-\ell \cos \theta &\quad \Rightarrow \quad {\dot {y}}_{\mathrm {pend} }=\ell {\dot {\theta }}\sin \theta \end{array}}}$

The generalized coordinates can be taken to be x and θ. The kinetic energy of the system is then

${\displaystyle T={\frac {1}{2}}M{\dot {x}}^{2}+{\frac {1}{2}}m\left({\dot {x}}_{\mathrm {pend} }^{2}+{\dot {y}}_{\mathrm {pend} }^{2}\right)}$

and the potential energy is

${\displaystyle V=mgy_{\mathrm {pend} }}$

giving the Lagrangian

${\displaystyle {\begin{array}{rcl}L&=&T-V\\&=&{\frac {1}{2}}M{\dot {x}}^{2}+{\frac {1}{2}}m\left[\left({\dot {x}}+\ell {\dot {\theta }}\cos \theta \right)^{2}+\left(\ell {\dot {\theta }}\sin \theta \right)^{2}\right]+mg\ell \cos \theta \\&=&{\frac {1}{2}}\left(M+m\right){\dot {x}}^{2}+m{\dot {x}}\ell {\dot {\theta }}\cos \theta +{\frac {1}{2}}m\ell ^{2}{\dot {\theta }}^{2}+mg\ell \cos \theta \end{array}}}$

Since x is absent from the Lagrangian, it is a cyclic coordinate. The conserved momentum is

${\displaystyle p_{x}={\frac {\partial L}{\partial {\dot {x}}}}=(M+m){\dot {x}}+m\ell {\dot {\theta }}\cos \theta \,.}$

and the Lagrange equation for the support coordinate x is

${\displaystyle (M+m){\ddot {x}}+m\ell {\ddot {\theta }}\cos \theta -m\ell {\dot {\theta }}^{2}\sin \theta =0}$

The Lagrange equation for the angle θ is

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left[m({\dot {x}}\ell \cos \theta +\ell ^{2}{\dot {\theta }})\right]+m\ell ({\dot {x}}{\dot {\theta }}+g)\sin \theta =0;}$

and simplifying

${\displaystyle {\ddot {\theta }}+{\frac {\ddot {x}}{\ell }}\cos \theta +{\frac {g}{\ell }}\sin \theta =0.}$

These equations may look quite complicated, but finding them with Newton's laws would have required carefully identifying all forces, which would have been much more laborious and prone to errors. By considering limit cases, the correctness of this system can be verified: For example, ${\displaystyle {\ddot {x}}\to 0}$ should give the equations of motion for a simple pendulum that is at rest in some inertial frame, while ${\displaystyle {\ddot {\theta }}\to 0}$ should give the equations for a pendulum in a constantly accelerating system, etc. Furthermore, it is trivial to obtain the results numerically, given suitable starting conditions and a chosen time step, by stepping through the results iteratively.

Two-body central force problem

Two bodies of masses m₁ and m₂ with position vectors r₁ and r₂ are in orbit about each other due to an attractive central potential V. We may write down the Lagrangian in terms of the position coordinates as they are, but it is an established procedure to convert the two-body problem into a one-body problem as follows. Introduce the Jacobi coordinates; the separation of the bodies r = r₂ − r₁ and the location of the center of mass R = (m₁r₁ + m₂r₂)/(m₁ + m₂). The Lagrangian is then

${\displaystyle L=\underbrace {{\frac {1}{2}}M{\dot {\mathbf {R} }}^{2}} _{L_{\text{cm}}}+\underbrace {{\frac {1}{2}}\mu {\dot {\mathbf {r} }}^{2}-V(|\mathbf {r} |)} _{L_{\text{rel}}}}$

where M = m₁ + m₂ is the total mass, μ = m₁m₂/(m₁ + m₂) is the reduced mass, and V the potential of the radial force, which depends only on the magnitude of the separation |r| = |r₂ − r₁|. The Lagrangian splits into a center-of-mass term L_cm and a relative motion term L_rel.

The Euler–Lagrange equation for R is simply

${\displaystyle M{\ddot {\mathbf {R} }}=0\,,}$

which states the center of mass moves in a straight line at constant velocity.

Since the relative motion only depends on the magnitude of the separation, it is ideal to use polar coordinates (r, θ) and take r = |r|,

${\displaystyle L_{\text{rel}}={\frac {1}{2}}\mu ({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2})-V(r)\,,}$

so θ is an ignorable coordinate with the corresponding conserved (angular) momentum

${\displaystyle p_{\theta }={\frac {\partial L_{\text{rel}}}{\partial {\dot {\theta }}}}=\mu r^{2}{\dot {\theta }}=\ell \,.}$

The radial coordinate r and angular velocity dθ/dt can vary with time, but only in such a way that ℓ is constant. The Lagrange equation for r is

${\displaystyle \mu r{\dot {\theta }}^{2}-{\frac {dV}{dr}}=\mu {\ddot {r}}\,.}$

This equation is identical to the radial equation obtained using Newton's laws in a co-rotating reference frame, that is, a frame rotating with the reduced mass so it appears stationary. Eliminating the angular velocity dθ/dt from this radial equation,

${\displaystyle \mu {\ddot {r}}=-{\frac {\mathrm {d} V}{\mathrm {d} r}}+{\frac {\ell ^{2}}{\mu r^{3}}}\,.}$

which is the equation of motion for a one-dimensional problem in which a particle of mass μ is subjected to the inward central force − dV/dr and a second outward force, called in this context the centrifugal force

${\displaystyle F_{\mathrm {cf} }=\mu r{\dot {\theta }}^{2}={\frac {\ell ^{2}}{\mu r^{3}}}\,.}$

Of course, if one remains entirely within the one-dimensional formulation, ℓ enters only as some imposed parameter of the external outward force, and its interpretation as angular momentum depends upon the more general two-dimensional problem from which the one-dimensional problem originated.

If one arrives at this equation using Newtonian mechanics in a co-rotating frame, the interpretation is evident as the centrifugal force in that frame due to the rotation of the frame itself. If one arrives at this equation directly by using the generalized coordinates (r, θ) and simply following the Lagrangian formulation without thinking about frames at all, the interpretation is that the centrifugal force is an outgrowth of using polar coordinates. As Hildebrand says:

"Since such quantities are not true physical forces, they are often called inertia forces. Their presence or absence depends, not upon the particular problem at hand, but upon the coordinate system chosen."

In particular, if Cartesian coordinates are chosen, the centrifugal force disappears, and the formulation involves only the central force itself, which provides the centripetal force for a curved motion.

This viewpoint, that fictitious forces originate in the choice of coordinates, often is expressed by users of the Lagrangian method. This view arises naturally in the Lagrangian approach, because the frame of reference is (possibly unconsciously) selected by the choice of coordinates. For example, see for a comparison of Lagrangians in an inertial and in a noninertial frame of reference. Unfortunately, this usage of "inertial force" conflicts with the Newtonian idea of an inertial force. In the Newtonian view, an inertial force originates in the acceleration of the frame of observation (the fact that it is not an inertial frame of reference), not in the choice of coordinate system. To keep matters clear, it is safest to refer to the Lagrangian inertial forces as generalized inertial forces, to distinguish them from the Newtonian vector inertial forces. That is, one should avoid following Hildebrand when he says (p. 155) "we deal always with generalized forces, velocities accelerations, and momenta. For brevity, the adjective "generalized" will be omitted frequently."

It is known that the Lagrangian of a system is not unique. Within the Lagrangian formalism the Newtonian fictitious forces can be identified by the existence of alternative Lagrangians in which the fictitious forces disappear, sometimes found by exploiting the symmetry of the system.

Electromagnetism

A test particle is a particle whose mass and charge are assumed to be so small that its effect on external system is insignificant. It is often a hypothetical simplified point particle with no properties other than mass and charge. Real particles like electrons and up quarks are more complex and have additional terms in their Lagrangians.

The Lagrangian for a charged particle with electrical charge q, interacting with an electromagnetic field, is the prototypical example of a velocity-dependent potential. The electric scalar potential ϕ = ϕ(r, t) and magnetic vector potential A = A(r, t) are defined from the electric field E = E(r, t) and magnetic field B = B(r, t) as follows;

${\displaystyle \mathbf {E} =-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}\,,\quad \mathbf {B} =\nabla \times \mathbf {A} \,.}$

The Lagrangian of a massive charged test particle in an electromagnetic field is

${\displaystyle L={\frac {m}{2}}{\dot {\mathbf {r} }}^{2}-q\phi +q{\dot {\mathbf {r} }}\cdot \mathbf {A} \,,}$

which produces the Lorentz force law

${\displaystyle m{\ddot {\mathbf {r} }}=q\mathbf {E} +q{\dot {\mathbf {r} }}\times \mathbf {B} \,.}$

An interesting detail in this example is the generalized momentum conjugate to r is the ordinary momentum plus a contribution from the A field,

${\displaystyle \mathbf {p} ={\frac {\partial L}{\partial {\dot {\mathbf {r} }}}}=m{\dot {\mathbf {r} }}+q\mathbf {A} \,.}$

If r is cyclic, which happens if the ϕ and A fields are uniform (independent of position), then this expression for p given here is the conserved momentum, while the usual quantity mv is not. This relation is also used in the minimal coupling prescription in quantum mechanics and quantum field theory.

Extensions to include non-conservative forces

Dissipation (i.e. non-conservative systems) can also be treated with an effective Lagrangian formulated by a certain doubling of the degrees of freedom.

In a more general formulation, the forces could be both conservative and viscous. If an appropriate transformation can be found from the F_i, Rayleigh suggests using a dissipation function, D, of the following form:

${\displaystyle D={\frac {1}{2}}\sum _{j=1}^{m}\sum _{k=1}^{m}C_{jk}{\dot {q}}_{j}{\dot {q}}_{k}\,.}$

where C_jk are constants that are related to the damping coefficients in the physical system, though not necessarily equal to them. If D is defined this way, then

${\displaystyle Q_{j}=-{\frac {\partial V}{\partial q_{j}}}-{\frac {\partial D}{\partial {\dot {q}}_{j}}}}$

and

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)-{\frac {\partial L}{\partial q_{j}}}+{\frac {\partial D}{\partial {\dot {q}}_{j}}}=0\,.}$

Other contexts and formulations

The ideas in Lagrangian mechanics have numerous applications in other areas of physics, and can adopt generalized results from the calculus of variations.

Alternative formulations of classical mechanics

A closely related formulation of classical mechanics is Hamiltonian mechanics. The Hamiltonian is defined by

${\displaystyle H=\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-L\,.}$

and can be obtained by performing a Legendre transformation on the Lagrangian, which introduces new variables canonically conjugate to the original variables. For example, given a set of generalized coordinates, the variables canonically conjugate are the generalized momenta. This doubles the number of variables, but makes differential equations first order. The Hamiltonian is a particularly ubiquitous quantity in quantum mechanics.

Routhian mechanics is a hybrid formulation of Lagrangian and Hamiltonian mechanics, which is not often used in practice but an efficient formulation for cyclic coordinates.

Momentum space formulation

The Euler–Lagrange equations can also be formulated in terms of the generalized momenta rather than generalized coordinates. Performing a Legendre transformation on the generalized coordinate Lagrangian L(q, dq/dt, t) obtains the generalized momenta Lagrangian L′(p, dp/dt, t) in terms of the original Lagrangian, as well the EL equations in terms of the generalized momenta. Both Lagrangians contain the same information, and either can be used to solve for the motion of the system. In practice generalized coordinates are more convenient to use and interpret than generalized momenta.

Higher derivatives of generalized coordinates

There is no reason to restrict the derivatives of generalized coordinates to first order only. It is possible to derive modified EL equations for a Lagrangian containing higher order derivatives.

Optics

Lagrangian mechanics can be applied to geometrical optics, by applying variational principles to rays of light in a medium, and solving the EL equations gives the equations of the paths the light rays follow.

Relativistic formulation

Lagrangian mechanics can be formulated in special relativity and general relativity. Some features of Lagrangian mechanics are retained in the relativistic theories but difficulties quickly appear in other respects. In particular, the EL equations take the same form, and the connection between cyclic coordinates and conserved momenta still applies, however the Lagrangian must be modified and is not simply the kinetic minus the potential energy of a particle. Also, it is not straightforward to handle multiparticle systems in a manifestly covariant way, it may be possible if a particular frame of reference is singled out.

Quantum mechanics

In quantum mechanics, 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.

In 1948, Feynman discovered the path integral formulation extending the principle of least action to quantum mechanics for electrons and photons. In this formulation, particles travel every possible path between the initial and final states; the probability of a specific final state is obtained by summing over all possible trajectories leading to it. In the classical regime, the path integral formulation cleanly reproduces Hamilton's principle, and Fermat's principle in optics.

Classical field theory

In Lagrangian mechanics, the generalized coordinates form a discrete set of variables that define the configuration of a system. In classical field theory, the physical system is not a set of discrete particles, but rather a continuous field ϕ(r, t) defined over a region of 3d space. Associated with the field is a Lagrangian density

${\displaystyle {\mathcal {L}}(\phi ,\nabla \phi ,\partial \phi /\partial t,\mathbf {r} ,t)}$

defined in terms of the field and its space and time derivatives at a location r and time t. Analogous to the particle case, for non-relativistic applications the Lagrangian density is also the kinetic energy density of the field, minus its potential energy density (this is not true in general, and the Lagrangian density has to be "reverse engineered"). The Lagrangian is then the volume integral of the Lagrangian density over 3d space

${\displaystyle L(t)=\int {\mathcal {L}}\,\mathrm {d} ^{3}\mathbf {r} }$

where d³r is a 3d differential volume element. The Lagrangian is a function of time since the Lagrangian density has implicit space dependence via the fields, and may have explicit spatial dependence, but these are removed in the integral, leaving only time in as the variable for the Lagrangian.

Noether's theorem

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

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.

Joseph-Louis Lagrange

From Wikipedia, the free encyclopedia

Joseph-Louis Lagrange
Joseph-Louis (Giuseppe Luigi), comte de Lagrange
Born	Giuseppe Lodovico Lagrangia 25 January 1736 Turin, Piedmont-Sardinia
Died	10 April 1813 (aged 77) Paris, France
Residence	Piedmont France Prussia
Citizenship	Piedmont-Sardinia French Empire
Alma mater	University of Turin
Known for	Analytical mechanics Celestial mechanics Mathematical analysis Number theory Pisano period
Scientific career
Fields	Mathematics Mathematical physics
Institutions	École Normale École Polytechnique
Academic advisors	Leonhard Euler (epistolary correspondent) Giovanni Battista Beccaria
Notable students	Joseph Fourier Giovanni Plana Siméon Poisson
Influenced	Évariste Galois

Joseph-Louis Lagrange (/ləˈɡrɑːndʒ/ or /ləˈɡreɪndʒ/; French: [lagʁɑ̃ʒ]; born Giuseppe Luigi Lagrangia or Giuseppe Ludovico De la Grange Tournier, 25 January 1736 – 10 April 1813; also reported as Giuseppe Luigi Lagrange or Lagrangia) was an Italian Enlightenment Era mathematician and astronomer. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.

In 1766, on the recommendation of Euler and d'Alembert, Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, Prussia, where he stayed for over twenty years, producing volumes of work and winning several prizes of the French Academy of Sciences. Lagrange's treatise on analytical mechanics (Mécanique analytique, 4. ed., 2 vols. Paris: Gauthier-Villars et fils, 1788–89), written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since Newton and formed a basis for the development of mathematical physics in the nineteenth century.

In 1787, at age 51, he moved from Berlin to Paris and became a member of the French Academy of Sciences. He remained in France until the end of his life. He was significantly involved in the decimalisation in Revolutionary France, became the first professor of analysis at the École Polytechnique upon its opening in 1794, was a founding member of the Bureau des Longitudes, and became Senator in 1799.

Scientific contribution

Lagrange was one of the creators of the calculus of variations, deriving the Euler–Lagrange equations for extrema of functionals. He also extended the method to take into account possible constraints, arriving at the method of Lagrange multipliers. Lagrange invented the method of solving differential equations known as variation of parameters, applied differential calculus to the theory of probabilities and attained notable work on the solution of equations. He proved that every natural number is a sum of four squares. His treatise Theorie des fonctions analytiques laid some of the foundations of group theory, anticipating Galois. In calculus, Lagrange developed a novel approach to interpolation and Taylor series. He studied the three-body problem for the Earth, Sun and Moon (1764) and the movement of Jupiter's satellites (1766), and in 1772 found the special-case solutions to this problem that yield what are now known as Lagrangian points. But above all, he is best known for his work on mechanics, where he transformed Newtonian mechanics into a branch of analysis, Lagrangian mechanics as it is now called, and presented the so-called mechanical "principles" as simple results of the variational calculus.

Biography

In appearance he was of medium height, and slightly formed, with pale blue eyes and a colourless complexion. In character he was nervous and timid, he detested controversy, and to avoid it willingly allowed others to take the credit for what he had himself done.

He always thought out the subject of his papers before he began to compose them, and usually wrote them straight off without a single erasure or correction.

W.W. Rouse Ball

Early years

Born as Giuseppe Lodovico Lagrangia, Lagrange was of Italian and French descent. His paternal great-grandfather was a French army officer who had moved to Turin, the de facto capital of the kingdom of Piedmont-Sardinia at Lagrange's time, and married an Italian; so did his grandfather and his father. His mother was from the countryside of Turin. He was raised as a Roman Catholic (but later on became an agnostic).

His father, who had charge of the king's military chest and was Treasurer of the Office of Public Works and Fortifications in Turin, should have maintained a good social position and wealth, but before his son grew up he had lost most of his property in speculations. A career as a lawyer was planned out for Lagrange by his father, and certainly Lagrange seems to have accepted this willingly. He studied at the University of Turin and his favourite subject was classical Latin. At first he had no great enthusiasm for mathematics, finding Greek geometry rather dull.

It was not until he was seventeen that he showed any taste for mathematics – his interest in the subject being first excited by a paper by Edmond Halley which he came across by accident. Alone and unaided he threw himself into mathematical studies; at the end of a year's incessant toil he was already an accomplished mathematician. Charles Emmanuel III appointed Lagrange to serve as the "Sostituto del Maestro di Matematica" (mathematics assistant professor) at the Royal Military Academy of the Theory and Practice of Artillery in 1755, where he taught courses in calculus and mechanics to support the Piedmontese army's early adoption of the ballistics theories of Benjamin Robins and Leonhard Euler. In that capacity, Lagrange was the first to teach calculus in an engineering school. According to Alessandro Papacino D'Antoni, the academy's military commander and famous artillery theorist, Lagrange unfortunately proved to be a problematic professor with his oblivious teaching style, abstract reasoning, and impatience with artillery and fortification-engineering applications. In this Academy one of his students was François Daviet de Foncenex.

Variational calculus

Lagrange is one of the founders of the calculus of variations. Starting in 1754, he worked on the problem of tautochrone, discovering a method of maximising and minimising functionals in a way similar to finding extrema of functions. Lagrange wrote several letters to Leonhard Euler between 1754 and 1756 describing his results. He outlined his "δ-algorithm", leading to the Euler–Lagrange equations of variational calculus and considerably simplifying Euler's earlier analysis. Lagrange also applied his ideas to problems of classical mechanics, generalising the results of Euler and Maupertuis.

Euler was very impressed with Lagrange's results. It has been stated that "with characteristic courtesy he withheld a paper he had previously written, which covered some of the same ground, in order that the young Italian might have time to complete his work, and claim the undisputed invention of the new calculus"; however, this chivalric view has been disputed. Lagrange published his method in two memoirs of the Turin Society in 1762 and 1773.

Miscellanea Taurinensia

In 1758, with the aid of his pupils (mainly Daviet de Foncenex), Lagrange established a society, which was subsequently incorporated as the Turin Academy of Sciences, and most of his early writings are to be found in the five volumes of its transactions, usually known as the Miscellanea Taurinensia. Many of these are elaborate papers. The first volume contains a paper on the theory of the propagation of sound; in this he indicates a mistake made by Newton, obtains the general differential equation for the motion, and integrates it for motion in a straight line. This volume also contains the complete solution of the problem of a string vibrating transversely; in this paper he points out a lack of generality in the solutions previously given by Brook Taylor, D'Alembert, and Euler, and arrives at the conclusion that the form of the curve at any time t is given by the equation ${\displaystyle y=a\sin(mx)\sin(nt)\,}$. The article concludes with a masterly discussion of echoes, beats, and compound sounds. Other articles in this volume are on recurring series, probabilities, and the calculus of variations.

The second volume contains a long paper embodying the results of several papers in the first volume on the theory and notation of the calculus of variations; and he illustrates its use by deducing the principle of least action, and by solutions of various problems in dynamics.

The third volume includes the solution of several dynamical problems by means of the calculus of variations; some papers on the integral calculus; a solution of Fermat's problem mentioned above: given an integer n which is not a perfect square, to find a number x such that x²n + 1 is a perfect square; and the general differential equations of motion for three bodies moving under their mutual attractions.

The next work he produced was in 1764 on the libration of the Moon, and an explanation as to why the same face was always turned to the earth, a problem which he treated by the aid of virtual work. His solution is especially interesting as containing the germ of the idea of generalised equations of motion, equations which he first formally proved in 1780.

Berlin

Already in 1756, Euler and Maupertuis, seeing his mathematical talent, tried to persuade him to come to Berlin, but Lagrange had no such intention and shyly refused the offer. In 1765, d'Alembert interceded on Lagrange's behalf with Frederick of Prussia and by letter, asked him to leave Turin for a considerably more prestigious position in Berlin. Lagrange again turned down the offer, responding that

It seems to me that Berlin would not be at all suitable for me while M.Euler is there.

In 1766, Euler left Berlin for Saint Petersburg, and Frederick himself wrote to Lagrange expressing the wish of "the greatest king in Europe" to have "the greatest mathematician in Europe" resident at his court. Lagrange was finally persuaded and he spent the next twenty years in Prussia, where he produced not only the long series of papers published in the Berlin and Turin transactions, but also his monumental work, the Mécanique analytique. In 1767, he married his cousin Vittoria Conti.

Lagrange was a favourite of the king, who used frequently to discourse to him on the advantages of perfect regularity of life. The lesson went home, and thenceforth Lagrange studied his mind and body as though they were machines, and found by experiment the exact amount of work which he was able to do without breaking down. Every night he set himself a definite task for the next day, and on completing any branch of a subject he wrote a short analysis to see what points in the demonstrations or in the subject-matter were capable of improvement. He always thought out the subject of his papers before he began to compose them, and usually wrote them straight off without a single erasure or correction.

Nonetheless, during his years in Berlin, Lagrange's health was rather poor on many occasions, and that of his wife Vittoria was even worse. She died in 1783 after years of illness and Lagrange was very depressed. In 1786, Frederick II died, and the climate of Berlin became rather trying for Lagrange.

Paris

In 1786, following Frederick's death, Lagrange received similar invitations from states including Spain and Naples, and he accepted the offer of Louis XVI to move to Paris. In France he was received with every mark of distinction and special apartments in the Louvre were prepared for his reception, and he became a member of the French Academy of Sciences, which later became part of the Institut de France (1795). At the beginning of his residence in Paris he was seized with an attack of melancholy, and even the printed copy of his Mécanique on which he had worked for a quarter of a century lay for more than two years unopened on his desk. Curiosity as to the results of the French revolution first stirred him out of his lethargy, a curiosity which soon turned to alarm as the revolution developed.

It was about the same time, 1792, that the unaccountable sadness of his life and his timidity moved the compassion of 24-year-old Renée-Françoise-Adélaïde Le Monnier, daughter of his friend, the astronomer Pierre Charles Le Monnier. She insisted on marrying him, and proved a devoted wife to whom he became warmly attached.

In September 1793, the Reign of Terror began. Under intervention of Antoine Lavoisier, who himself was by then already thrown out of the Academy along with many other scholars, Lagrange was specifically exempted by name in the decree of October 1793 that ordered all foreigners to leave France. On 4 May 1794, Lavoisier and 27 other tax farmers were arrested and sentenced to death and guillotined on the afternoon after the trial. Lagrange said on the death of Lavoisier:

It took only a moment to cause this head to fall and a hundred years will not suffice to produce its like.

Though Lagrange had been preparing to escape from France while there was yet time, he was never in any danger; different revolutionary governments (and at a later time, Napoleon) loaded him with honours and distinctions. This luckiness or safety may to some extent be due to his life attitude he expressed many years before: "I believe that, in general, one of the first principles of every wise man is to conform strictly to the laws of the country in which he is living, even when they are unreasonable". A striking testimony to the respect in which he was held was shown in 1796 when the French commissary in Italy was ordered to attend in full state on Lagrange's father, and tender the congratulations of the republic on the achievements of his son, who "had done honor to all mankind by his genius, and whom it was the special glory of Piedmont to have produced." It may be added that Napoleon, when he attained power, warmly encouraged scientific studies in France, and was a liberal benefactor of them. Appointed senator in 1799, he was the first signer of the Sénatus-consulte which in 1802 annexed his fatherland Piedmont to France. He acquired French citizenship in consequence. The French claimed he was a French mathematician, but the Italians continued to claim him as Italian.

Units of measurement

Lagrange was considerably involved in the process of making new standard units of measurement in the 1790s. He was offered the presidency of the Commission for the reform of weights and measures (la Commission des Poids et Mesures) when he was preparing to escape. And after Lavoisier's death in 1794, it was largely owing to Lagrange's influence that the final choice of the unit system of metre and kilogram was settled and the decimal subdivision was finally accepted by the commission of 1799. Lagrange was also one of the founding members of the Bureau des Longitudes in 1795.

École Normale

In 1795, Lagrange was appointed to a mathematical chair at the newly established École Normale, which enjoyed only a brief existence of four months. His lectures there were quite elementary, and contain nothing of any special importance, but they were published because the professors had to "pledge themselves to the representatives of the people and to each other neither to read nor to repeat from memory," and the discourses were ordered to be taken down in shorthand to enable the deputies to see how the professors acquitted themselves.

École Polytechnique

In 1794, Lagrange was appointed professor of the École Polytechnique; and his lectures there, described by mathematicians who had the good fortune to be able to attend them, were almost perfect both in form and matter. Beginning with the merest elements, he led his hearers on until, almost unknown to themselves, they were themselves extending the bounds of the subject: above all he impressed on his pupils the advantage of always using general methods expressed in a symmetrical notation.

But Lagrange does not seem to have been a successful teacher. Fourier, who attended his lectures in 1795, wrote:

his voice is very feeble, at least in that he does not become heated; he has a very marked Italian accent and pronounces the s like z [...] The students, of whom the majority are incapable of appreciating him, give him little welcome, but the professeurs make amends for it.

Late years

Lagrange's tomb in the crypt of the Panthéon

In 1810, Lagrange commenced a thorough revision of the Mécanique analytique, but he was able to complete only about two-thirds of it before his death at Paris in 1813, in 128 rue du Faubourg Saint-Honoré. Napoleon honoured him with the Grand Croix of the Ordre Impérial de la Réunion just two days before he died. He was buried that same year in the Panthéon in Paris. The inscription on his tomb reads in translation:

JOSEPH LOUIS LAGRANGE. Senator. Count of the Empire. Grand Officer of the Legion of Honour. Grand Cross of the Imperial Order of the Reunion. Member of the Institute and the Bureau of Longitude. Born in Turin on 25 January 1736. Died in Paris on 10 April 1813.

Work in Berlin

Lagrange was extremely active scientifically during twenty years he spent in Berlin. Not only did he produce his Mécanique analytique, but he contributed between one and two hundred papers to the Academy of Turin, the Berlin Academy, and the French Academy. Some of these are really treatises, and all without exception are of a high order of excellence. Except for a short time when he was ill he produced on average about one paper a month. Of these, note the following as amongst the most important.

First, his contributions to the fourth and fifth volumes, 1766–1773, of the Miscellanea Taurinensia; of which the most important was the one in 1771, in which he discussed how numerous astronomical observations should be combined so as to give the most probable result. And later, his contributions to the first two volumes, 1784–1785, of the transactions of the Turin Academy; to the first of which he contributed a paper on the pressure exerted by fluids in motion, and to the second an article on integration by infinite series, and the kind of problems for which it is suitable.

Most of the papers sent to Paris were on astronomical questions, and among these including his paper on the Jovian system in 1766, his essay on the problem of three bodies in 1772, his work on the secular equation of the Moon in 1773, and his treatise on cometary perturbations in 1778. These were all written on subjects proposed by the Académie française, and in each case the prize was awarded to him.

Lagrangian mechanics

Between 1772 and 1788, Lagrange re-formulated Classical/Newtonian mechanics to simplify formulas and ease calculations. These mechanics are called Lagrangian mechanics.

Algebra

The greater number of his papers during this time were, however, contributed to the Prussian Academy of Sciences. Several of them deal with questions in algebra.

• His discussion of representations of integers by quadratic forms (1769) and by more general algebraic forms (1770).
• His tract on the Theory of Elimination, 1770.
• Lagrange's theorem that the order of a subgroup H of a group G must divide the order of G.
• His papers of 1770 and 1771 on the general process for solving an algebraic equation of any degree via the Lagrange resolvents. This method fails to give a general formula for solutions of an equation of degree five and higher, because the auxiliary equation involved has higher degree than the original one. The significance of this method is that it exhibits the already known formulas for solving equations of second, third, and fourth degrees as manifestations of a single principle, and was foundational in Galois theory. The complete solution of a binomial equation of any degree is also treated in these papers.
• In 1773, Lagrange considered a functional determinant of order 3, a special case of a Jacobian. He also proved the expression for the volume of a tetrahedron with one of the vertices at the origin as the one sixth of the absolute value of the determinant formed by the coordinates of the other three vertices.

Number theory

Several of his early papers also deal with questions of number theory.

• Lagrange (1766–1769) was the first European to prove that Pell's equation x² − ny² = 1 has a nontrivial solution in the integers for any non-square natural number n.
• He proved the theorem, stated by Bachet without justification, that every positive integer is the sum of four squares, 1770.
• He proved Wilson's theorem that (for any integer n > 1): n is a prime if and only if (n − 1)! + 1 is a multiple of n, 1771.
• His papers of 1773, 1775, and 1777 gave demonstrations of several results enunciated by Fermat, and not previously proved.
• His Recherches d'Arithmétique of 1775 developed a general theory of binary quadratic forms to handle the general problem of when an integer is representable by the form ax² + by² + cxy.
• He made contributions to the theory of continued fractions.

Other mathematical work

There are also numerous articles on various points of analytical geometry. In two of them, written rather later, in 1792 and 1793, he reduced the equations of the quadrics (or conicoids) to their canonical forms.

During the years from 1772 to 1785, he contributed a long series of papers which created the science of partial differential equations. A large part of these results were collected in the second edition of Euler's integral calculus which was published in 1794.

Astronomy

Lastly, there are numerous papers on problems in astronomy. Of these the most important are the following:

• Attempting to solve the general three-body problem, with the consequent discovery of the two constant-pattern solutions, collinear and equilateral, 1772. Those solutions were later seen to explain what are now known as the Lagrangian points.
• On the attraction of ellipsoids, 1773: this is founded on Maclaurin's work.
• On the secular equation of the Moon, 1773; also noticeable for the earliest introduction of the idea of the potential. The potential of a body at any point is the sum of the mass of every element of the body when divided by its distance from the point. Lagrange showed that if the potential of a body at an external point were known, the attraction in any direction could be at once found. The theory of the potential was elaborated in a paper sent to Berlin in 1777.
• On the motion of the nodes of a planet's orbit, 1774.
• On the stability of the planetary orbits, 1776.
• Two papers in which the method of determining the orbit of a comet from three observations is completely worked out, 1778 and 1783: this has not indeed proved practically available, but his system of calculating the perturbations by means of mechanical quadratures has formed the basis of most subsequent researches on the subject.
• His determination of the secular and periodic variations of the elements of the planets, 1781–1784: the upper limits assigned for these agree closely with those obtained later by Le Verrier, and Lagrange proceeded as far as the knowledge then possessed of the masses of the planets permitted.
• Three papers on the method of interpolation, 1783, 1792 and 1793: the part of finite differences dealing therewith is now in the same stage as that in which Lagrange left it.

Mécanique analytique

Over and above these various papers he composed his great treatise, the Mécanique analytique. In this he lays down the law of virtual work, and from that one fundamental principle, by the aid of the calculus of variations, deduces the whole of mechanics, both of solids and fluids.

The object of the book is to show that the subject is implicitly included in a single principle, and to give general formulae from which any particular result can be obtained. The method of generalised co-ordinates by which he obtained this result is perhaps the most brilliant result of his analysis. Instead of following the motion of each individual part of a material system, as D'Alembert and Euler had done, he showed that, if we determine its configuration by a sufficient number of variables whose number is the same as that of the degrees of freedom possessed by the system, then the kinetic and potential energies of the system can be expressed in terms of those variables, and the differential equations of motion thence deduced by simple differentiation. For example, in dynamics of a rigid system he replaces the consideration of the particular problem by the general equation, which is now usually written in the form

${\displaystyle {\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {\theta }}}}-{\frac {\partial T}{\partial \theta }}+{\frac {\partial V}{\partial \theta }}=0,}$

where T represents the kinetic energy and V represents the potential energy of the system. He then presented what we now know as the method of Lagrange multipliers—though this is not the first time that method was published—as a means to solve this equation. Amongst other minor theorems here given it may suffice to mention the proposition that the kinetic energy imparted by the given impulses to a material system under given constraints is a maximum, and the principle of least action. All the analysis is so elegant that Sir William Rowan Hamilton said the work could be described only as a scientific poem. Lagrange remarked that mechanics was really a branch of pure mathematics analogous to a geometry of four dimensions, namely, the time and the three coordinates of the point in space; and it is said that he prided himself that from the beginning to the end of the work there was not a single diagram. At first no printer could be found who would publish the book; but Legendre at last persuaded a Paris firm to undertake it, and it was issued under the supervision of Laplace, Cousin, Legendre (editor) and Condorcet in 1788.

Work in France

Differential calculus and calculus of variations

Lagrange's lectures on the differential calculus at École Polytechnique form the basis of his treatise Théorie des fonctions analytiques, which was published in 1797. This work is the extension of an idea contained in a paper he had sent to the Berlin papers in 1772, and its object is to substitute for the differential calculus a group of theorems based on the development of algebraic functions in series, relying in particular on the principle of the generality of algebra.

A somewhat similar method had been previously used by John Landen in the Residual Analysis, published in London in 1758. Lagrange believed that he could thus get rid of those difficulties, connected with the use of infinitely large and infinitely small quantities, to which philosophers objected in the usual treatment of the differential calculus. The book is divided into three parts: of these, the first treats of the general theory of functions, and gives an algebraic proof of Taylor's theorem, the validity of which is, however, open to question; the second deals with applications to geometry; and the third with applications to mechanics.

Another treatise on the same lines was his Leçons sur le calcul des fonctions, issued in 1804, with the second edition in 1806. It is in this book that Lagrange formulated his celebrated method of Lagrange multipliers, in the context of problems of variational calculus with integral constraints. These works devoted to differential calculus and calculus of variations may be considered as the starting point for the researches of Cauchy, Jacobi, and Weierstrass.

Infinitesimals

At a later period Lagrange fully embraced the use of infinitesimals in preference to founding the differential calculus on the study of algebraic forms; and in the preface to the second edition of the Mécanique Analytique, which was issued in 1811, he justifies the employment of infinitesimals, and concludes by saying that:

When we have grasped the spirit of the infinitesimal method, and have verified the exactness of its results either by the geometrical method of prime and ultimate ratios, or by the analytical method of derived functions, we may employ infinitely small quantities as a sure and valuable means of shortening and simplifying our proofs.

Number theory

His Résolution des équations numériques, published in 1798, was also the fruit of his lectures at École Polytechnique. There he gives the method of approximating to the real roots of an equation by means of continued fractions, and enunciates several other theorems. In a note at the end he shows how Fermat's little theorem, that is

${\displaystyle a^{p-1}-1\equiv 0{\pmod {p}}}$

where p is a prime and a is prime to p, may be applied to give the complete algebraic solution of any binomial equation. He also here explains how the equation whose roots are the squares of the differences of the roots of the original equation may be used so as to give considerable information as to the position and nature of those roots.

Celestial mechanics

The theory of the planetary motions had formed the subject of some of the most remarkable of Lagrange's Berlin papers. In 1806 the subject was reopened by Poisson, who, in a paper read before the French Academy, showed that Lagrange's formulae led to certain limits for the stability of the orbits. Lagrange, who was present, now discussed the whole subject afresh, and in a letter communicated to the Academy in 1808 explained how, by the variation of arbitrary constants, the periodical and secular inequalities of any system of mutually interacting bodies could be determined.

Prizes and distinctions

Euler proposed Lagrange for election to the Berlin Academy and he was elected on 2 September 1756. He was elected a Fellow of the Royal Society of Edinburgh in 1790, a Fellow of the Royal Society and a foreign member of the Royal Swedish Academy of Sciences in 1806. In 1808, Napoleon made Lagrange a Grand Officer of the Legion of Honour and a Count of the Empire. He was awarded the Grand Croix of the Ordre Impérial de la Réunion in 1813, a week before his death in Paris.

Lagrange was awarded the 1764 prize of the French Academy of Sciences for his memoir on the libration of the Moon. In 1766 the Academy proposed a problem of the motion of the satellites of Jupiter, and the prize again was awarded to Lagrange. He also shared or won the prizes of 1772, 1774, and 1778.

Lagrange is one of the 72 prominent French scientists who were commemorated on plaques at the first stage of the Eiffel Tower when it first opened. Rue Lagrange in the 5th Arrondissement in Paris is named after him. In Turin, the street where the house of his birth still stands is named via Lagrange. The lunar crater Lagrange also bears his name.