Routhian mechanics

Last updated
Edward John Routh, 1831-1907 Edward J Routh.jpg
Edward John Routh, 1831–1907

In classical mechanics, Routh's procedure or Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the Lagrangian and Hamiltonian functions. Routhian mechanics is equivalent to Lagrangian mechanics and Hamiltonian mechanics, and introduces no new physics. It offers an alternative way to solve mechanical problems.

Contents

Definitions

The Routhian, like the Hamiltonian, can be obtained from a Legendre transform of the Lagrangian, and has a similar mathematical form to the Hamiltonian, but is not exactly the same. The difference between the Lagrangian, Hamiltonian, and Routhian functions are their variables. For a given set of generalized coordinates representing the degrees of freedom in the system, the Lagrangian is a function of the coordinates and velocities, while the Hamiltonian is a function of the coordinates and momenta.

The Routhian differs from these functions in that some coordinates are chosen to have corresponding generalized velocities, the rest to have corresponding generalized momenta. This choice is arbitrary, and can be done to simplify the problem. It also has the consequence that the Routhian equations are exactly the Hamiltonian equations for some coordinates and corresponding momenta, and the Lagrangian equations for the rest of the coordinates and their velocities. In each case the Lagrangian and Hamiltonian functions are replaced by a single function, the Routhian. The full set thus has the advantages of both sets of equations, with the convenience of splitting one set of coordinates to the Hamilton equations, and the rest to the Lagrangian equations.

In the case of Lagrangian mechanics, the generalized coordinates q1, q2, ... and the corresponding velocities dq1/dt, dq2/dt, ..., and possibly time [nb 1] t, enter the Lagrangian,

where the overdots denote time derivatives.

In Hamiltonian mechanics, the generalized coordinates q1, q2, ... and the corresponding generalized momenta p1, p2, ..., and possibly time, enter the Hamiltonian,

where the second equation is the definition of the generalized momentum pi corresponding to the coordinate qi (partial derivatives are denoted using ). The velocities dqi/dt are expressed as functions of their corresponding momenta by inverting their defining relation. In this context, pi is said to be the momentum "canonically conjugate" to qi.

The Routhian is intermediate between L and H; some coordinates q1, q2, ..., qn are chosen to have corresponding generalized momenta p1, p2, ..., pn, the rest of the coordinates ζ1, ζ2, ..., ζs to have generalized velocities 1/dt, 2/dt, ..., s/dt, and time may appear explicitly; [1] [2]

Routhian (n + s degrees of freedom)

where again the generalized velocity dqi/dt is to be expressed as a function of generalized momentum pi via its defining relation. The choice of which n coordinates are to have corresponding momenta, out of the n + s coordinates, is arbitrary.

The above is used by Landau and Lifshitz, and Goldstein. Some authors may define the Routhian to be the negative of the above definition. [3]

Given the length of the general definition, a more compact notation is to use boldface for tuples (or vectors) of the variables, thus q = (q1, q2, ..., qn), ζ = (ζ1, ζ2, ..., ζs), p = (p1, p2, ..., pn), and dζ/dt = (1/dt, 2/dt, ..., s/dt), so that

where · is the dot product defined on the tuples, for the specific example appearing here:

Equations of motion

For reference, the Euler-Lagrange equations for s degrees of freedom are a set of s coupled second order ordinary differential equations in the coordinates

where j = 1, 2, ..., s, and the Hamiltonian equations for n degrees of freedom are a set of 2n coupled first order ordinary differential equations in the coordinates and momenta

Below, the Routhian equations of motion are obtained in two ways, in the process other useful derivatives are found that can be used elsewhere.

Two degrees of freedom

Consider the case of a system with two degrees of freedom, q and ζ, with generalized velocities dq/dt and /dt, and the Lagrangian is time-dependent. (The generalization to any number of degrees of freedom follows exactly the same procedure as with two). [4] The Lagrangian of the system will have the form

The differential of L is

Now change variables, from the set (q, ζ, dq/dt, /dt) to (q, ζ, p, /dt), simply switching the velocity dq/dt to the momentum p. This change of variables in the differentials is the Legendre transformation. The differential of the new function to replace L will be a sum of differentials in dq, , dp, d(/dt), and dt. Using the definition of generalized momentum and Lagrange's equation for the coordinate q:

we have

and to replace pd(dq/dt) by (dq/dt)dp, recall the product rule for differentials, [nb 2] and substitute

to obtain the differential of a new function in terms of the new set of variables:

Introducing the Routhian

where again the velocity dq/dt is a function of the momentum p, we have

but from the above definition, the differential of the Routhian is

Comparing the coefficients of the differentials dq, , dp, d(/dt), and dt, the results are Hamilton's equations for the coordinate q,

and Lagrange's equation for the coordinate ζ

which follow from

and taking the total time derivative of the second equation and equating to the first. Notice the Routhian replaces the Hamiltonian and Lagrangian functions in all the equations of motion.

The remaining equation states the partial time derivatives of L and R are negatives

Any number of degrees of freedom

For n + s coordinates as defined above, with Routhian

the equations of motion can be derived by a Legendre transformation of this Routhian as in the previous section, but another way is to simply take the partial derivatives of R with respect to the coordinates qi and ζj, momenta pi, and velocities j/dt, where i = 1, 2, ..., n, and j = 1, 2, ..., s. The derivatives are

The first two are identically the Hamiltonian equations. Equating the total time derivative of the fourth set of equations with the third (for each value of j) gives the Lagrangian equations. The fifth is just the same relation between time partial derivatives as before. To summarize [5]

Routhian equations of motion (n + s degrees of freedom)

The total number of equations is 2n + s, there are 2n Hamiltonian equations plus s Lagrange equations.

Energy

Since the Lagrangian has the same units as energy, the units of the Routhian are also energy. In SI units this is the Joule.

Taking the total time derivative of the Lagrangian leads to the general result

If the Lagrangian is independent of time, the partial time derivative of the Lagrangian is zero, L/∂t = 0, so the quantity under the total time derivative in brackets must be a constant, it is the total energy of the system [6]

(If there are external fields interacting with the constituents of the system, they can vary throughout space but not time). This expression requires the partial derivatives of L with respect to all the velocities dqi/dt and j/dt. Under the same condition of R being time independent, the energy in terms of the Routhian is a little simpler, substituting the definition of R and the partial derivatives of R with respect to the velocities j/dt,

Notice only the partial derivatives of R with respect to the velocities j/dt are needed. In the case that s = 0 and the Routhian is explicitly time-independent, then E = R, that is, the Routhian equals the energy of the system. The same expression for R in when s = 0 is also the Hamiltonian, so in all E = R = H.

If the Routhian has explicit time dependence, the total energy of the system is not constant. The general result is

which can be derived from the total time derivative of R in the same way as for L.

Cyclic coordinates

Often the Routhian approach may offer no advantage, but one notable case where this is useful is when a system has cyclic coordinates (also called "ignorable coordinates"), by definition those coordinates which do not appear in the original Lagrangian. The Lagrangian equations are powerful results, used frequently in theory and practice, since the equations of motion in the coordinates are easy to set up. However, if cyclic coordinates occur there will still be equations to solve for all the coordinates, including the cyclic coordinates despite their absence in the Lagrangian. The Hamiltonian equations are useful theoretical results, but less useful in practice because coordinates and momenta are related together in the solutions - after solving the equations the coordinates and momenta must be eliminated from each other. Nevertheless, the Hamiltonian equations are perfectly suited to cyclic coordinates because the equations in the cyclic coordinates trivially vanish, leaving only the equations in the non cyclic coordinates.

The Routhian approach has the best of both approaches, because cyclic coordinates can be split off to the Hamiltonian equations and eliminated, leaving behind the non cyclic coordinates to be solved from the Lagrangian equations. Overall fewer equations need to be solved compared to the Lagrangian approach.

The Routhian formulation is useful for systems with cyclic coordinates, because by definition those coordinates do not enter L, and hence R. The corresponding partial derivatives of L and R with respect to those coordinates are zero, which equates to the corresponding generalized momenta reducing to constants. To make this concrete, if the qi are all cyclic coordinates, and the ζj are all non cyclic, then

where the αi are constants. With these constants substituted into the Routhian, R is a function of only the non cyclic coordinates and velocities (and in general time also)

The 2n Hamiltonian equation in the cyclic coordinates automatically vanishes,

and the s Lagrangian equations are in the non cyclic coordinates

Thus the problem has been reduced to solving the Lagrangian equations in the non cyclic coordinates, with the advantage of the Hamiltonian equations cleanly removing the cyclic coordinates. Using those solutions, the equations for can be integrated to compute .

If we are interested in how the cyclic coordinates change with time, the equations for the generalized velocities corresponding to the cyclic coordinates can be integrated.

Examples

Routh's procedure does not guarantee the equations of motion will be simple, however it will lead to fewer equations.

Central potential in spherical coordinates

One general class of mechanical systems with cyclic coordinates are those with central potentials, because potentials of this form only have dependence on radial separations and no dependence on angles.

Consider a particle of mass m under the influence of a central potential V(r) in spherical polar coordinates (r, θ, φ)

Notice φ is cyclic, because it does not appear in the Lagrangian. The momentum conjugate to φ is the constant

in which r and /dt can vary with time, but the angular momentum pφ is constant. The Routhian can be taken to be

We can solve for r and θ using Lagrange's equations, and do not need to solve for φ since it is eliminated by Hamiltonian's equations. The r equation is

and the θ equation is

The Routhian approach has obtained two coupled nonlinear equations. By contrast the Lagrangian approach leads to three nonlinear coupled equations, mixing in the first and second time derivatives of φ in all of them, despite its absence from the Lagrangian.

The r equation is

the θ equation is

the φ equation is

Symmetric mechanical systems

Spherical pendulum

Spherical pendulum: angles and velocities Spherical pendulum Lagrangian mechanics.svg
Spherical pendulum: angles and velocities

Consider the spherical pendulum, a mass m (known as a "pendulum bob") attached to a rigid rod of length l of negligible mass, subject to a local gravitational field g. The system rotates with angular velocity /dt which is not constant. The angle between the rod and vertical is θ and is not constant.

The Lagrangian is [nb 3]

and φ is the cyclic coordinate for the system with constant momentum

which again is physically the angular momentum of the system about the vertical. The angle θ and angular velocity /dt vary with time, but the angular momentum is constant. The Routhian is

The θ equation is found from the Lagrangian equations

or simplifying by introducing the constants

gives

This equation resembles the simple nonlinear pendulum equation, because it can swing through the vertical axis, with an additional term to account for the rotation about the vertical axis (the constant a is related to the angular momentum pφ).

Applying the Lagrangian approach there are two nonlinear coupled equations to solve.

The θ equation is

and the φ equation is

Heavy symmetrical top

Heavy symmetric top in terms of the Euler angles Heavy symmetric top euler angles.svg
Heavy symmetric top in terms of the Euler angles

The heavy symmetrical top of mass M has Lagrangian [7] [8]

where ψ, φ, θ are the Euler angles, θ is the angle between the vertical z-axis and the top's z-axis, ψ is the rotation of the top about its own z-axis, and φ the azimuthal of the top's z-axis around the vertical z-axis. The principal moments of inertia are I1 about the top's own x axis, I2 about the top's own y axes, and I3 about the top's own z-axis. Since the top is symmetric about its z-axis, I1 = I2. Here the simple relation for local gravitational potential energy V = Mglcosθ is used where g is the acceleration due to gravity, and the centre of mass of the top is a distance l from its tip along its z-axis.

The angles ψ, φ are cyclic. The constant momenta are the angular momenta of the top about its axis and its precession about the vertical, respectively:

From these, eliminating /dt:

we have

and to eliminate /dt, substitute this result into pψ and solve for /dt to find

The Routhian can be taken to be

and since

we have

The first term is constant, and can be ignored since only the derivatives of R will enter the equations of motion. The simplified Routhian, without loss of information, is thus

The equation of motion for θ is, by direct calculation,

or by introducing the constants

a simpler form of the equation is obtained

Although the equation is highly nonlinear, there is only one equation to solve for, it was obtained directly, and the cyclic coordinates are not involved.

By contrast, the Lagrangian approach leads to three nonlinear coupled equations to solve, despite the absence of the coordinates ψ and φ in the Lagrangian.

The θ equation is

the ψ equation is

and the φ equation is

Velocity-dependent potentials

Classical charged particle in a uniform magnetic field

Classical charged particle in uniform B field, using cylindrical coordinates. Top: If the radial coordinate r and angular velocity dth/dt vary, the trajectory is a helicoid with varying radius but uniform motion in the z direction. Bottom: Constant r and dth/dt means a helicoid with constant radius. Charged particle in uniform B field.svg
Classical charged particle in uniform B field, using cylindrical coordinates. Top: If the radial coordinate r and angular velocity /dt vary, the trajectory is a helicoid with varying radius but uniform motion in the z direction. Bottom: Constant r and /dt means a helicoid with constant radius.

Consider a classical charged particle of mass m and electric charge q in a static (time-independent) uniform (constant throughout space) magnetic field B. [9] The Lagrangian for a charged particle in a general electromagnetic field given by the magnetic potential A and electric potential is

It is convenient to use cylindrical coordinates (r, θ, z), so that

In this case of no electric field, the electric potential is zero, , and we can choose the axial gauge for the magnetic potential

and the Lagrangian is

Notice this potential has an effectively cylindrical symmetry (although it also has angular velocity dependence), since the only spatial dependence is on the radial length from an imaginary cylinder axis.

There are two cyclic coordinates, θ and z. The canonical momenta conjugate to θ and z are the constants

so the velocities are

The angular momentum about the z axis is notpθ, but the quantity mr2/dt, which is not conserved due to the contribution from the magnetic field. The canonical momentum pθ is the conserved quantity. It is still the case that pz is the linear or translational momentum along the z axis, which is also conserved.

The radial component r and angular velocity /dt can vary with time, but pθ is constant, and since pz is constant it follows dz/dt is constant. The Routhian can take the form

where in the last line, the pz2/2m term is a constant and can be ignored without loss of continuity. The Hamiltonian equations for θ and z automatically vanish and do not need to be solved for. The Lagrangian equation in r

is by direct calculation

which after collecting terms is

and simplifying further by introducing the constants

the differential equation is

To see how z changes with time, integrate the momenta expression for pz above

where cz is an arbitrary constant, the initial value of z to be specified in the initial conditions.

The motion of the particle in this system is helicoidal, with the axial motion uniform (constant) but the radial and angular components varying in a spiral according to the equation of motion derived above. The initial conditions on r, dr/dt, θ, /dt, will determine if the trajectory of the particle has a constant r or varying r. If initially r is nonzero but dr/dt = 0, while θ and /dt are arbitrary, then the initial velocity of the particle has no radial component, r is constant, so the motion will be in a perfect helix. If r is constant, the angular velocity is also constant according to the conserved pθ.

With the Lagrangian approach, the equation for r would include /dt which has to be eliminated, and there would be equations for θ and z to solve for.

The r equation is

the θ equation is

and the z equation is

The z equation is trivial to integrate, but the r and θ equations are not, in any case the time derivatives are mixed in all the equations and must be eliminated.

See also

Footnotes

  1. The coordinates are functions of time, so the Lagrangian always has implicit time-dependence via the coordinates. If the Lagrangian changes with time irrespective of the coordinates, usually due to some time-dependent potential, then the Lagrangian is said to have "explicit" time-dependence. Similarly for the Hamiltonian and Routhian functions.
  2. For two functions u and v, the differential of the product is d(uv) = udv + vdu.
  3. The potential energy is actually
    but since the first term is constant, it can be ignored in the Lagrangian (and Routhian) which only depend on derivatives of coordinates and velocities. Subtracting this from the kinetic energy means a plus sign in the Lagrangian, not minus.

Notes

Related Research Articles

<span class="mw-page-title-main">Laplace's equation</span> Second-order partial differential equation

In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as

<span class="mw-page-title-main">Navier–Stokes equations</span> Equations describing the motion of viscous fluid substances

The Navier–Stokes equations are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

<span class="mw-page-title-main">Equations of motion</span> Equations that describe the behavior of a physical system

In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition.

<span class="mw-page-title-main">Hamiltonian mechanics</span> Formulation of classical mechanics using momenta

Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena.

In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the 18th century and onward, after Newtonian mechanics. Since Newtonian mechanics considers vector quantities of motion, particularly accelerations, momenta, forces, of the constituents of the system, an alternative name for the mechanics governed by Newton's laws and Euler's laws is vectorial mechanics.

In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state. The generalized velocities are the time derivatives of the generalized coordinates of the system. The adjective "generalized" distinguishes these parameters from the traditional use of the term "coordinate" to refer to Cartesian coordinates.

<span class="mw-page-title-main">Spherical pendulum</span>

In physics, a spherical pendulum is a higher dimensional analogue of the pendulum. It consists of a mass m moving without friction on the surface of a sphere. The only forces acting on the mass are the reaction from the sphere and gravity.

In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics.

The Kuramoto model, first proposed by Yoshiki Kuramoto, is a mathematical model used in describing synchronization. More specifically, it is a model for the behavior of a large set of coupled oscillators. Its formulation was motivated by the behavior of systems of chemical and biological oscillators, and it has found widespread applications in areas such as neuroscience and oscillating flame dynamics. Kuramoto was quite surprised when the behavior of some physical systems, namely coupled arrays of Josephson junctions, followed his model.

<span class="mw-page-title-main">Axial multipole moments</span>

Axial multipole moments are a series expansion of the electric potential of a charge distribution localized close to the origin along one Cartesian axis, denoted here as the z-axis. However, the axial multipole expansion can also be applied to any potential or field that varies inversely with the distance to the source, i.e., as . For clarity, we first illustrate the expansion for a single point charge, then generalize to an arbitrary charge density localized to the z-axis.

In fluid mechanics, potential vorticity (PV) is a quantity which is proportional to the dot product of vorticity and stratification. This quantity, following a parcel of air or water, can only be changed by diabatic or frictional processes. It is a useful concept for understanding the generation of vorticity in cyclogenesis, especially along the polar front, and in analyzing flow in the ocean.

In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.

In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

<span class="mw-page-title-main">Mild-slope equation</span> Physics phenomenon and formula

In fluid dynamics, the mild-slope equation describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used in coastal engineering to compute the wave-field changes near harbours and coasts.

<span class="mw-page-title-main">Lagrangian mechanics</span> Formulation of classical mechanics

In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the Turin Academy of Science in 1760 culminating in his 1788 grand opus, Mécanique analytique.

In general relativity, the Vaidya metric describes the non-empty external spacetime of a spherically symmetric and nonrotating star which is either emitting or absorbing null dusts. It is named after the Indian physicist Prahalad Chunnilal Vaidya and constitutes the simplest non-static generalization of the non-radiative Schwarzschild solution to Einstein's field equation, and therefore is also called the "radiating(shining) Schwarzschild metric".

Calculations in the Newman–Penrose (NP) formalism of general relativity normally begin with the construction of a complex null tetrad, where is a pair of real null vectors and is a pair of complex null vectors. These tetrad vectors respect the following normalization and metric conditions assuming the spacetime signature

In physics, the distorted Schwarzschild metric is the metric of a standard/isolated Schwarzschild spacetime exposed in external fields. In numerical simulation, the Schwarzschild metric can be distorted by almost arbitrary kinds of external energy–momentum distribution. However, in exact analysis, the mature method to distort the standard Schwarzschild metric is restricted to the framework of Weyl metrics.

Conformastatic spacetimes refer to a special class of static solutions to Einstein's equation in general relativity.

References