Lagrangian mechanics

The Lagrangian formalism is in physics one introduced by Joseph Louis Lagrange in 1788 formulation of classical mechanics, in which the dynamics of a system is described by a single scalar function, the Lagrange function. The formalism is (in contrast to Newtonian mechanics, the a priori valid only in inertial frames ) in accelerated reference systems valid. The Lagrangian formalism is invariant under coordinate transformations. From the Lagrangian, the equations of motion using the Euler -Lagrange equations of the calculus of variations can be determined from the principle of least action. This approach simplifies many physical problems as opposed to the Newtonian formulation of the laws of motion in the Lagrangian formalism constraints are relatively easy to take into account by explicitly calculating the constraining forces or the appropriate choice of generalized coordinates.

For systems with a generalized potential and holonomic constraints is the Lagrangian

Wherein the kinetic energy and the potential energy of the system under consideration, respectively. We distinguish so-called Lagrange equations of the first and second kind in the narrower sense, under the Lagrangian formalism and the Lagrange equations but the second kind, which are often referred to simply as Lagrange equations:

Here are generalized coordinates and their time derivatives.

Lagrange equations of the first and second type

With the Lagrange equations of the first kind can be explicitly calculate the constraint forces. They are equivalent to the equations arising from the D' Alembert principle. We consider point particles in the position vectors, whose coordinates are constrained by independent ( holonomic ) constraints of the form ( with an explicit time dependence was allowed). Characterized the positions of the particles are limited to a -dimensional manifold (the number of degrees of freedom ).

The constraint forces are perpendicular to this manifold and can therefore be represented by a linear combination of the gradients:

If one assumes that the external forces can be derived from a potential, one can write the equation of motion as follows ( Lagrange equation 1 type ):

Are the masses of the point-like particles is the potential energy. This, together with the constraints are of 3N s independent equations for the 3N and the coordinates for s Lagrange multipliers. Thus, the solution of the equation system is unique.

Note: This only holonomic constraints were treated. The formalism can, however, also be applied to constraints of the form that follow, for example, in non- holonomic constraints between the velocities of the particles. This constraint equations can be in contrast to holonomic constraints do not pose as a complete differential of a function, that is, between the coefficient functions does not apply.

In the case of holonomic constraints, one can introduce new coordinates that include them by implication, so-called generalized coordinates. With the kinetic energy

And potential forces

(which are also expressed by generalized coordinates and are then called generalized forces - they do not necessarily have the dimension of a force) can also write the equations of motion

Or with the Lagrangian ( Lagrange equation second kind ):

If, as in this case only of a potential derivable forces ( potential forces), it is called by conservative forces.

Note: Sometimes the generalized forces can be written by a speed-dependent generalized potential in the following form

Even then, there are the equations of motion

With the Lagrangian:

The system is then not more conservative in the usual sense. An example is the case of the electromagnetic field ( see below).

Sometimes one has yet non-conservative forces, so that write the equations:

An example are systems with non- holonomic constraints ( see above) or frictional forces.

Derivation from the Hamilton 's principle

The Lagrange equations of the second kind arise as so-called Euler -Lagrange equations of a variational problem and provide the equations of motion when the Lagrangian is given. Follow from variation of the action integral formed with the Lagrangian in Hamilton 's principle. This purpose we consider all possible trajectories in the space of generalized coordinates between fixed starting and ending points. We consider the change in the action integral with variation of the trajectories

The Hamiltonian principle states that the action integral is for the classic track stationary under variation of the trajectories:

An approximation to first order is for an ordinary function f (x, y)


To first order, the variation of the integral is therefore to

Now one performs a partial integration into the term that contains the derivative with respect to time:

Here is used that

, since the start and end point are recorded. Therefore subject to the boundary terms

This ultimately results

Now that occurs as a factor of the total integral and is arbitrary, the integral can only disappear with the variational principle, if the integrand itself vanishes. The following are the Lagrange equations or Lagrange equations of the second kind ( the Euler -Lagrange equations of the variational problem considered here ):

For each generalized coordinate ( and the associated generalized velocity) there is such an equation. The Lagrange equations form a system of ordinary differential equations of second order with respect to the time derivative. How many differential equations which are, in effect, you only know if the number of degrees of freedom of the "system " were calculated.

Cyclic variables and symmetry

If the Lagrangian does not depend on one coordinate, but only by the associated speed it is called cyclic, cyclic or cyclic coordinate variable. The conjugated to the cyclic variable pulse

Is a conserved quantity: its value does not change during the motion, as will be shown the same. If the Lagrangian does not depend on, applies

However, it follows from the Euler-Lagrange equation, the time derivative of the corresponding conjugated pulse disappears, and thus it is constant in time:

More generally belongs to the Noether theorem to each continuous symmetry of the action of a conserved quantity. In a cyclical variable effect is invariant under the displacement of an arbitrary constant,

Extension to fields

In the field theory, the equation of motion from Hamilton's principle for fields that yields to

Where the observed field and the Lagrangian density.

One can write this in shorthand as

With the so- defined variation dissipation.

The Lagrangian formalism is also the starting point of many formulations of quantum field theory.

Relativistic mechanics

In relativistic mechanics, the Lagrangian of a free particle are derived from Hamilton's principle by a relativistic scalar is assumed for the effect of the simplest case:

Which is proportional to the proper time relativistic line element and a constant factor was chosen.

The Lagrangian of a free particle is here no longer identical with the kinetic energy ( sometimes you therefore also speaks of kinetic energy supplement T in the Lagrangian ). Is the relativistic kinetic energy of a body with rest mass and velocity without constraints

While for the Lagrangian, the kinetic energy supplement

Is relevant. The Lagrangian for a particle in a potential V is then given by

For one - particle system, the Lagrange function is associated with the generalized coordinates

Wherein the number of degrees of freedom and the number of holonomic constraints is.

For small velocities one can develop the root to first order:

The zeroth order of development is a constant, the negative rest energy. Since the Lagrange equations are invariant under addition of a constant to the Lagrangian, one can neglect the constant first term, and we obtain the classical kinetic energy again:

Connection with path integrals in quantum mechanics

Richard Feynman has also used this approach first consistently for the derivation of the equations of quantum mechanics. In classical physics, there are the above Lagrange equations from the requirement that the action integral is stationary. In Feynman's path integral formalism is the quantum mechanical probability amplitude that a system between the initial and final conditions strikes a specific path, is proportional to the action integral. Paths in the vicinity of the classical pathway, for which the variation of disappears, it usually provide the main contributions because the contributions add up with almost the same phase factors in their environment.


Mass in the harmonic potential ( conservative)

A mass is connected by two springs with spring constant and fixed boundary conditions. Prerequisite to define the problem in the Lagrange formalism is putting up the Lagrangian, by setting up the terms for kinetic energy and potential energy:

The Lagrangian is therefore:

The Lagrange function is again used for the analytical description of the physical problem in the Euler-Lagrange equation, which then leads to equations corresponding to the equations of motion in the Newtonian mechanics. In this example, the generalized coordinate, the Euler -Lagrange equation

And from this

Lead to the equation of motion of the system:

The general solution of this differential equation is, the time, the angular frequency. The constant amplitude and phase can be determined from the initial conditions.

Charge in the electromagnetic field ( non- conservative)

A point charge with mass is moving in the electromagnetic field. The generalized coordinates correspond to the Cartesian coordinates in three space dimensions.

The fields ( electric and magnetic fields ) are determined by the scalar potential and the vector potential:

The kinetic energy of the particle is classic:

The " potential " is here, however, depending on speed, it is called therefore as shown above by a generalized potential:

Thus, the Lagrangian of a charged particle in an electromagnetic field:

The Euler -Lagrange equations leads to the equation of motion on the right side of the Lorentz force is:

Mass of the drum ( non- conservative)

The axis of a winding drum is driven by a torque M. The mass of the load is m, the moment of inertia of the drum is J. The radius of the drum is r.

The following relationship exists between the coordinates x and φ:

The kinetic energy is

The virtual work of the impressed forces is

It finally follows the equation of motion

The resolution of this equation for the angular acceleration results

Atwood 's machine (method of the first kind )

In an Atwood 's machine considering two point masses in the gravitational field of the earth, which are suspended over a pulley at height h and connected by a rope of length l. The constraint condition in this case is:

If a rope is taken into account, which is on the roll (roll radius r), then the result is:

The potential energy V is calculated as:

For the gradient is obtained

This leads to the system of the first type Lagrange equations:

This can be dissolved and replaced by, for example, for known initial conditions:

Particles in free fall ( general relativity )

In the general theory of relativity freely falling particles world lines run through the longest time: between two ( sufficiently close to each other ) events and goes on an on-board clock on the world line of a free-falling particles more time than at all other world lines by these events. Be a monotonically increasing along the path running parameters, we obtain the elapsed time to

With the Lagrangian

In this case, the component functions to the metric ( both spatial and time components ). We expect simplicity in measurement systems, in which the speed of light is dimensionless and has a value, and use the Einstein summation convention.

The conjugate momentum is to

And are the Euler -Lagrange equations

Do we use here as an abbreviation, the Christoffel symbol

Then the world 's longest line length proves to be straight: the direction of the tangent to the world line

Does not change with parallel displacement along the world line

The parameterization is not fixed. Do we have such over them, that the tangent vector is everywhere the same length, then the tangent vector is constant and goes as it travels the world line into itself. It satisfies the geodesic equation

This is the general relativistic form of the equation of motion of a freely falling particle. Gravitation is fully taken into account in the.