Generalized coordinates

Physical quantities in generalized coordinates

Kinetic energy

The total kinetic energy of the system is the energy of the system's motion, defined as

${\displaystyle T=\frac{1}{2}\sum _{k=1}^N{m}_k{\dot{\mathbf{r}}}_k\cdot {\dot{\mathbf{r}}}_k,}$

in which · is the dot product. The kinetic energy is a function only of the velocities v_k, not the coordinates r_k themselves. By contrast an important observation is

${\displaystyle {\dot{\mathbf{r}}}_k\cdot {\dot{\mathbf{r}}}_k=\sum _{i,j=1}^n\left(\frac{\mathrm{\partial }{\mathbf{r}}_k}{\mathrm{\partial }{q}_i}\cdot \frac{\mathrm{\partial }{\mathbf{r}}_k}{\mathrm{\partial }{q}_j}\right){\dot{q}}_i{\dot{q}}_j,}$

which illustrates the kinetic energy is in general a function of the generalized velocities, coordinates, and time if the constraints also vary with time, so T = T(q, dq/dt, t).

In the case the constraints on the particles are time-independent, then all partial derivatives with respect to time are zero, and the kinetic energy is a homogeneous function of degree 2 in the generalized velocities.

Still for the time-independent case, this expression is equivalent to taking the line element squared of the trajectory for particle k,

${\displaystyle d{s}_k^2=d{\mathbf{r}}_k\cdot d{\mathbf{r}}_k=\sum _{i,j=1}^n\left(\frac{\mathrm{\partial }{\mathbf{r}}_k}{\mathrm{\partial }{q}_i}\cdot \frac{\mathrm{\partial }{\mathbf{r}}_k}{\mathrm{\partial }{q}_j}\right)d{q}_{i}d{q}_j,}$

and dividing by the square differential in time, dt², to obtain the velocity squared of particle k. Thus for time-independent constraints it is sufficient to know the line element to quickly obtain the kinetic energy of particles and hence the Lagrangian.

It is instructive to see the various cases of polar coordinates in 2D and 3D, owing to their frequent appearance. In 2D polar coordinates (r, θ),

${\displaystyle {\left(\frac{ds}{dt}\right)}^2={\dot{r}}^2+r^2{\dot{\theta }}^2,}$

in 3D cylindrical coordinates (r, θ, z),

${\displaystyle {\left(\frac{ds}{dt}\right)}^2={\dot{r}}^2+r^2{\dot{\theta }}^2+{\dot{z}}^2,}$

in 3D spherical coordinates (r, θ, φ),

${\displaystyle {\left(\frac{ds}{dt}\right)}^2={\dot{r}}^2+r^2{\dot{\theta }}^2+r^2{\sin }^2\theta {\dot{\phi }}^2.}$

Generalized momentum

The generalized momentum "canonically conjugate to" the coordinate q_i is defined by

${\displaystyle p_i=\frac{\mathrm{\partial }L}{\mathrm{\partial }{\dot{q}}_i}.}$

If the Lagrangian L does not depend on some coordinate q_i, then it follows from the Euler–Lagrange equations that the corresponding generalized momentum will be a conserved quantity, because the time derivative is zero implying the momentum is a constant of the motion;

${\displaystyle \frac{\mathrm{\partial }L}{\mathrm{\partial }{q}_i}=\frac{d}{dt}\frac{\mathrm{\partial }L}{\mathrm{\partial }{\dot{q}}_i}={\dot{p}}_i=0.}$

Examples

Bead on a wire

For a bead sliding on a frictionless wire subject only to gravity in 2d space, the constraint on the bead can be stated in the form f (r) = 0, where the position of the bead can be written r = (x(s), y(s)), in which s is a parameter, the arc length s along the curve from some point on the wire. This is a suitable choice of generalized coordinate for the system. Only one coordinate is needed instead of two, because the position of the bead can be parameterized by one number, s, and the constraint equation connects the two coordinates x and y; either one is determined from the other. The constraint force is the reaction force the wire exerts on the bead to keep it on the wire, and the non-constraint applied force is gravity acting on the bead.

Suppose the wire changes its shape with time, by flexing. Then the constraint equation and position of the particle are respectively

${\displaystyle f(\mathbf{r},t)=0,\mathbf{r}=(x(s,t),y(s,t))}$

which now both depend on time t due to the changing coordinates as the wire changes its shape. Notice time appears implicitly via the coordinates and explicitly in the constraint equations.

Simple pendulum

The relationship between the use of generalized coordinates and Cartesian coordinates to characterize the movement of a mechanical system can be illustrated by considering the constrained dynamics of a simple pendulum.

A simple pendulum consists of a mass M hanging from a pivot point so that it is constrained to move on a circle of radius L. The position of the mass is defined by the coordinate vector r = (x, y) measured in the plane of the circle such that y is in the vertical direction. The coordinates x and y are related by the equation of the circle

${\displaystyle f(x,y)=x^2+y^2-L^2=0,}$

that constrains the movement of M. This equation also provides a constraint on the velocity components,

${\displaystyle \dot{f}(x,y)=2x\dot{x}+2y\dot{y}=0.}$

Now introduce the parameter θ, that defines the angular position of M from the vertical direction. It can be used to define the coordinates x and y, such that

${\displaystyle \mathbf{r}=(x,y)=(L\sin \theta ,-L\cos \theta ).}$

The use of θ to define the configuration of this system avoids the constraint provided by the equation of the circle.

Notice that the force of gravity acting on the mass m is formulated in the usual Cartesian coordinates,

${\displaystyle \mathbf{F}=(0,-mg),}$

where g is the acceleration due to gravity.

The virtual work of gravity on the mass m as it follows the trajectory r is given by

${\displaystyle \delta W=\mathbf{F}\cdot \delta \mathbf{r}.}$

The variation δr can be computed in terms of the coordinates x and y, or in terms of the parameter θ,

${\displaystyle \delta \mathbf{r}=(\delta x,\delta y)=(L\cos \theta ,L\sin \theta )\delta \theta .}$

Thus, the virtual work is given by

${\displaystyle \delta W=-mg\delta y=-mgL\sin (\theta )\delta \theta .}$

Notice that the coefficient of δy is the y-component of the applied force. In the same way, the coefficient of δθ is known as the generalized force along generalized coordinate θ, given by

${\displaystyle F_{\theta }=-mgL\sin \theta .}$

To complete the analysis consider the kinetic energy T of the mass, using the velocity,

${\displaystyle \mathbf{v}=(\dot{x},\dot{y})=(L\cos \theta ,L\sin \theta )\dot{\theta },}$

so,

${\displaystyle T=\frac{1}{2}m\mathbf{v}\cdot \mathbf{v}=\frac{1}{2}m({\dot{x}}^2+{\dot{y}}^2)=\frac{1}{2}m{L}^2{\dot{\theta }}^2.}$

D'Alembert's form of the principle of virtual work for the pendulum in terms of the coordinates x and y are given by,

${\displaystyle \frac{d}{dt}\frac{\mathrm{\partial }T}{\mathrm{\partial }\dot{x}}-\frac{\mathrm{\partial }T}{\mathrm{\partial }x}=F_x+\lambda \frac{\mathrm{\partial }f}{\mathrm{\partial }x},\frac{d}{dt}\frac{\mathrm{\partial }T}{\mathrm{\partial }\dot{y}}-\frac{\mathrm{\partial }T}{\mathrm{\partial }y}=F_y+\lambda \frac{\mathrm{\partial }f}{\mathrm{\partial }y}.}$

This yields the three equations

${\displaystyle m\ddot{x}=\lambda (2x),m\ddot{y}=-mg+\lambda (2y),x^2+y^2-L^2=0,}$

in the three unknowns, x, y and λ.

Using the parameter θ, those equations take the form

${\displaystyle \frac{d}{dt}\frac{\mathrm{\partial }T}{\mathrm{\partial }\dot{\theta }}-\frac{\mathrm{\partial }T}{\mathrm{\partial }\theta }=F_{\theta },}$

which becomes,

${\displaystyle m{L}^2\ddot{\theta }=-mgL\sin \theta ,}$

or

${\displaystyle \ddot{\theta }+\frac{g}{L}\sin \theta =0.}$

This formulation yields one equation because there is a single parameter and no constraint equation.

This shows that the parameter θ is a generalized coordinate that can be used in the same way as the Cartesian coordinates x and y to analyze the pendulum.

Double pendulum

The benefits of generalized coordinates become apparent with the analysis of a double pendulum. For the two masses m_i (i = 1, 2), let r_i = (x_i, y_i), i = 1, 2 define their two trajectories. These vectors satisfy the two constraint equations,

${\displaystyle f_1(x_1,y_1,x_2,y_2)={\mathbf{r}}_1\cdot {\mathbf{r}}_1-L_1^2=0}$

and

${\displaystyle f_2(x_1,y_1,x_2,y_2)=({\mathbf{r}}_2-{\mathbf{r}}_1)\cdot ({\mathbf{r}}_2-{\mathbf{r}}_1)-L_2^2=0.}$

The formulation of Lagrange's equations for this system yields six equations in the four Cartesian coordinates x_i, y_i (i = 1, 2) and the two Lagrange multipliers λ_i (i = 1, 2) that arise from the two constraint equations.

Now introduce the generalized coordinates θ_i (i = 1, 2) that define the angular position of each mass of the double pendulum from the vertical direction. In this case, we have

${\displaystyle {\mathbf{r}}_1=(L_1\sin {\theta }_1,-L_1\cos {\theta }_1),{\mathbf{r}}_2=(L_1\sin {\theta }_1,-L_1\cos {\theta }_1)+(L_2\sin {\theta }_2,-L_2\cos {\theta }_2).}$

The force of gravity acting on the masses is given by,

${\displaystyle {\mathbf{F}}_1=(0,-m_{1}g),{\mathbf{F}}_2=(0,-m_{2}g)}$

where g is the acceleration due to gravity. Therefore, the virtual work of gravity on the two masses as they follow the trajectories r_i (i = 1, 2) is given by

${\displaystyle \delta W={\mathbf{F}}_1\cdot \delta {\mathbf{r}}_1+{\mathbf{F}}_2\cdot \delta {\mathbf{r}}_2.}$

The variations δr_i (i = 1, 2) can be computed to be

${\displaystyle \delta {\mathbf{r}}_1=(L_1\cos {\theta }_1,L_1\sin {\theta }_1)\delta {\theta }_1,\delta {\mathbf{r}}_2=(L_1\cos {\theta }_1,L_1\sin {\theta }_1)\delta {\theta }_1+(L_2\cos {\theta }_2,L_2\sin {\theta }_2)\delta {\theta }_2}$

Thus, the virtual work is given by

${\displaystyle \delta W=-(m_1+m_2)g{L}_1\sin {\theta }_1\delta {\theta }_1-m_{2}g{L}_2\sin {\theta }_2\delta {\theta }_2,}$

and the generalized forces are

${\displaystyle F_{{\theta }_1}=-(m_1+m_2)g{L}_1\sin {\theta }_1,F_{{\theta }_2}=-m_{2}g{L}_2\sin {\theta }_2.}$

Compute the kinetic energy of this system to be

${\displaystyle T=\frac{1}{2}{m}_1{\mathbf{v}}_1\cdot {\mathbf{v}}_1+\frac{1}{2}{m}_2{\mathbf{v}}_2\cdot {\mathbf{v}}_2=\frac{1}{2}(m_1+m_2)L_1^2{\dot{\theta }}_1^2+\frac{1}{2}{m}_2{L}_2^2{\dot{\theta }}_2^2+m_2{L}_1{L}_2\cos ({\theta }_2-{\theta }_1){\dot{\theta }}_1{\dot{\theta }}_2.}$

Euler–Lagrange equation yield two equations in the unknown generalized coordinates θ_i (i = 1, 2) given by

${\displaystyle (m_1+m_2)L_1^2{\ddot{\theta }}_1+m_2{L}_1{L}_2{\ddot{\theta }}_2\cos ({\theta }_2-{\theta }_1)+m_2{L}_1{L}_2{\dot{{\theta }_2}}^2\sin ({\theta }_1-{\theta }_2)=-(m_1+m_2)g{L}_1\sin {\theta }_1,}$

and

${\displaystyle m_2{L}_2^2{\ddot{\theta }}_2+m_2{L}_1{L}_2{\ddot{\theta }}_1\cos ({\theta }_2-{\theta }_1)+m_2{L}_1{L}_2{\dot{{\theta }_1}}^2\sin ({\theta }_2-{\theta }_1)=-m_{2}g{L}_2\sin {\theta }_2.}$

The use of the generalized coordinates θ_i (i = 1, 2) provides an alternative to the Cartesian formulation of the dynamics of the double pendulum.

Spherical pendulum

For a 3D example, a spherical pendulum with constant length l free to swing in any angular direction subject to gravity, the constraint on the pendulum bob can be stated in the form

${\displaystyle f(\mathbf{r})=x^2+y^2+z^2-l^2=0,}$

where the position of the pendulum bob can be written

${\displaystyle \mathbf{r}=(x(\theta ,\varphi ),y(\theta ,\varphi ),z(\theta ,\varphi )),}$

in which (θ, φ) are the spherical polar angles because the bob moves in the surface of a sphere. The position r is measured along the suspension point to the bob, here treated as a point particle. A logical choice of generalized coordinates to describe the motion are the angles (θ, φ). Only two coordinates are needed instead of three, because the position of the bob can be parameterized by two numbers, and the constraint equation connects the three coordinates (x, y, z) so any one of them is determined from the other two.

Generalized coordinates and virtual work

The principle of virtual work states that if a system is in static equilibrium, the virtual work of the applied forces is zero for all virtual movements of the system from this state, that is, δW = 0 for any variation δr. When formulated in terms of generalized coordinates, this is equivalent to the requirement that the generalized forces for any virtual displacement are zero, that is F_i = 0.

Let the forces on the system be F_j (j = 1, 2, …, m) be applied to points with Cartesian coordinates r_j (j = 1, 2, …, m), then the virtual work generated by a virtual displacement from the equilibrium position is given by

${\displaystyle \delta W=\sum _{j=1}^m{\mathbf{F}}_j\cdot \delta {\mathbf{r}}_j.}$

where δr_j (j = 1, 2, …, m) denote the virtual displacements of each point in the body.

Now assume that each δr_j depends on the generalized coordinates q_i (i = 1, 2, …, n) then

${\displaystyle \delta {\mathbf{r}}_j=\frac{\mathrm{\partial }{\mathbf{r}}_j}{\mathrm{\partial }{q}_1}\delta q_1+\dots +\frac{\mathrm{\partial }{\mathbf{r}}_j}{\mathrm{\partial }{q}_n}\delta q_n,}$

and

${\displaystyle \delta W=\left(\sum _{j=1}^m{\mathbf{F}}_j\cdot \frac{\mathrm{\partial }{\mathbf{r}}_j}{\mathrm{\partial }{q}_1}\right)\delta q_1+\dots +\left(\sum _{j=1}^m{\mathbf{F}}_j\cdot \frac{\mathrm{\partial }{\mathbf{r}}_j}{\mathrm{\partial }{q}_n}\right)\delta q_n.}$

The n terms

${\displaystyle F_i=\sum _{j=1}^m{\mathbf{F}}_j\cdot \frac{\mathrm{\partial }{\mathbf{r}}_j}{\mathrm{\partial }{q}_i},i=1,\dots ,n,}$

are the generalized forces acting on the system. Kane shows that these generalized forces can also be formulated in terms of the ratio of time derivatives,

${\displaystyle F_i=\sum _{j=1}^m{\mathbf{F}}_j\cdot \frac{\mathrm{\partial }{\mathbf{v}}_j}{\mathrm{\partial }{\dot{q}}_i},i=1,\dots ,n,}$

where v_j is the velocity of the point of application of the force F_j.

In order for the virtual work to be zero for an arbitrary virtual displacement, each of the generalized forces must be zero, that is

${\displaystyle \delta W=0\Rightarrow F_i=0,i=1,\dots ,n.}$

Bibliography of cited references

• Ginsberg, Jerry H. (2008). Engineering dynamics (3rd ed.). Cambridge UK: Cambridge University Press. ISBN 978-0-521-88303-0.
• Goldstein, Herbert; Poole, Charles; Safko, John (2002). Classical Mechanics (3rd ed.). San Francisco, CA: Addison Wesley. ISBN 0-201-65702-3.
• Hand, Louis N.; Finch, Janet D. (1998). Analytical mechanics. Cambridge: Cambridge University Press. ISBN 978-0521575720.
• Kibble, T.W.B; Berkshire, F.H. (2004). Classical Mechanics (5th ed.). River Edge NJ: Imperial College Press. ISBN 1860944248.
• Landau, L. D.; Lifshitz, E.M. (1976). Mechanics (Third ed.). Oxford. ISBN 978-0750628969.{{cite book}}: CS1 maint: location missing publisher (link)
• Torby, Bruce (1984). "Energy Methods". Advanced Dynamics for Engineers. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing. ISBN 0-03-063366-4.