Chapter 9

Lagrangian Formulation

Theory, worked examples and 22 previous-year questions with full solutions — IIT JAM, GATE, JEST and CSIR NET-JRF, 2020 onwards.

Theory

9.1 Generalized coordinates and degrees of freedom

Newton’s laws, applied in Chapters 1 to 8, describe the motion of every particle by its Cartesian coordinates and require an explicit account of every force, including the forces of constraint exerted by strings, rods, surfaces and guides. For systems with many constraints this is laborious, and the unknown constraint forces are usually of no interest. The Lagrangian formulation removes them from the outset. The configuration of the system is described by the smallest set of independent variables that fixes it, and the equations of motion follow from a single scalar function, the Lagrangian, by a rule that has the same form in every set of coordinates.

(a) Coordinate systems and the kinetic energy

The position of a particle at a point \(P\) is fixed by its position vector \(\mathbf r\) or, equivalently, by its Cartesian coordinates \((x,y,z)\) with respect to the orthonormal basis \(\hat{\mathbf i},\hat{\mathbf j},\hat{\mathbf k}\) (Figure 9.1(a)). Then \(\mathbf r=x\hat{\mathbf i}+y\hat{\mathbf j}+z\hat{\mathbf k}\), the distance \(OP\) is \(r=\sqrt{x^2+y^2+z^2}\), and two points \(P_1(x_1,y_1,z_1)\) and \(P_2(x_2,y_2,z_2)\) are separated by \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\). Since \(d\mathbf r=dx\,\hat{\mathbf i}+dy\,\hat{\mathbf j}+dz\,\hat{\mathbf k}\), the square of an element of length is \(ds^2=dx^2+dy^2+dz^2\), the velocity is \(\mathbf v=\dot x\hat{\mathbf i}+\dot y\hat{\mathbf j}+\dot z\hat{\mathbf k}\), and the kinetic energy of a particle of mass \(m\) is

\[T=\frac12m\left(\dot x^2+\dot y^2+\dot z^2\right).\]

Figure 9.1 (a) Cartesian, (b) cylindrical and (c) spherical polar coordinates of a point.

In cylindrical coordinates \((r,\theta,z)\) (Figure 9.1(b)), \(x=r\cos\theta\), \(y=r\sin\theta\) and \(z=z\), so that \(\mathbf r=r\cos\theta\,\hat{\mathbf i}+r\sin\theta\,\hat{\mathbf j}+z\hat{\mathbf k}\). Differentiating with respect to time,

\[\mathbf v=(\dot r\cos\theta-r\dot\theta\sin\theta)\,\hat{\mathbf i}+(\dot r\sin\theta+r\dot\theta\cos\theta)\,\hat{\mathbf j}+\dot z\,\hat{\mathbf k}=\dot r\,\hat{\mathbf r}+r\dot\theta\,\hat{\boldsymbol\theta}+\dot z\,\hat{\mathbf z},\]

and the kinetic energy is

\[\boxed{T=\frac12m\left(\dot r^2+r^2\dot\theta^2+\dot z^2\right)}.\]

These coordinates suit problems with an axis of symmetry, in which there are one angular and two linear motions. Setting \(z=0\) gives the plane polar coordinates of Chapter 3, with \(T=\frac12m(\dot r^2+r^2\dot\theta^2)\).

In spherical polar coordinates \((r,\theta,\phi)\) (Figure 9.1(c)), \(x=r\sin\theta\cos\phi\), \(y=r\sin\theta\sin\phi\) and \(z=r\cos\theta\), where \(\theta\) is measured from the \(z\)-axis and \(\phi\) is the azimuth of the projection \(r\sin\theta\) on the \(xy\)-plane. Differentiating,

\[\begin{aligned}\dot x&=\dot r\sin\theta\cos\phi+r\dot\theta\cos\theta\cos\phi-r\dot\phi\sin\theta\sin\phi,\\ \dot y&=\dot r\sin\theta\sin\phi+r\dot\theta\cos\theta\sin\phi+r\dot\phi\sin\theta\cos\phi,\\ \dot z&=\dot r\cos\theta-r\dot\theta\sin\theta.\end{aligned}\]

The cross terms cancel on squaring and adding, so that \(\mathbf v=\dot r\,\hat{\mathbf r}+r\dot\theta\,\hat{\boldsymbol\theta}+r\sin\theta\,\dot\phi\,\hat{\boldsymbol\phi}\) and

\[\boxed{T=\frac12m\left(\dot r^2+r^2\dot\theta^2+r^2\sin^2\theta\,\dot\phi^2\right)}.\]

These coordinates suit problems with a centre of symmetry, in which there are two angular and one linear motion.

(b) Generalized coordinates and velocities

The coordinates used above are special cases of a general idea. Any set of independent quantities \(q_1,q_2,\dots,q_n\) that specifies the configuration of a system completely and uniquely is called a set of generalized coordinates, and their time derivatives \(\dot q_i=dq_i/dt\) are the generalized velocities. The generalized coordinates need not have the dimensions of length, nor the generalized velocities those of velocity. If \(q_i\) is a length \(r\), then \(\dot q_i=\dot r\) is a linear velocity; if \(q_i\) is an angle \(\theta\), then \(\dot q_i=\dot\theta\) is an angular velocity; and if a momentum \(p\) were chosen as a coordinate, its generalized velocity \(\dot p\) would have the dimensions of force.

More generally, let a point be specified by three single-valued functions of position \(u^1,u^2,u^3\), so that \(x=f_1(u^1,u^2,u^3)\), \(y=f_2(u^1,u^2,u^3)\) and \(z=f_3(u^1,u^2,u^3)\), with an inverse transformation \(u^k=F_k(x,y,z)\). Then \(\mathbf r=\mathbf r(u^1,u^2,u^3)\), and an element of displacement is

\[d\mathbf r=\sum_{i=1}^3\frac{\partial\mathbf r}{\partial u^i}\,du^i,\qquad ds^2=d\mathbf r\cdot d\mathbf r=\sum_{i=1}^3\sum_{j=1}^3\frac{\partial\mathbf r}{\partial u^i}\cdot\frac{\partial\mathbf r}{\partial u^j}\,du^i\,du^j.\]

Dividing by \(dt^2\) gives \(v^2\), and hence the kinetic energy, in the new coordinates. The coefficients \(\partial\mathbf r/\partial u^i\cdot\partial\mathbf r/\partial u^j\) form the metric of the coordinate system; for orthogonal coordinates only the diagonal terms survive, as in the cylindrical and spherical cases above.

Example 9.1 A particle of mass \(m\) moves in the plane \(z=0\), and its coordinates are given in terms of two angles by \(x=a\sin\theta_1+a\sin\theta_2\) and \(y=a\cos\theta_1+a\cos\theta_2\). Find its kinetic energy in terms of \(\theta_1\) and \(\theta_2\).

Solution. Here \(\mathbf r=(a\sin\theta_1+a\sin\theta_2)\,\hat{\mathbf i}+(a\cos\theta_1+a\cos\theta_2)\,\hat{\mathbf j}\), so

\[d\mathbf r=\frac{\partial\mathbf r}{\partial\theta_1}d\theta_1+\frac{\partial\mathbf r}{\partial\theta_2}d\theta_2=(a\cos\theta_1\,d\theta_1+a\cos\theta_2\,d\theta_2)\,\hat{\mathbf i}-(a\sin\theta_1\,d\theta_1+a\sin\theta_2\,d\theta_2)\,\hat{\mathbf j}.\]

The velocity is therefore \(\mathbf v=a(\dot\theta_1\cos\theta_1+\dot\theta_2\cos\theta_2)\,\hat{\mathbf i}-a(\dot\theta_1\sin\theta_1+\dot\theta_2\sin\theta_2)\,\hat{\mathbf j}\), and

\[\begin{aligned}T=\frac12mv^2&=\frac12ma^2\left[\dot\theta_1^2+\dot\theta_2^2+2\dot\theta_1\dot\theta_2(\cos\theta_1\cos\theta_2+\sin\theta_1\sin\theta_2)\right]\\&=\frac12ma^2\left[\dot\theta_1^2+\dot\theta_2^2+2\dot\theta_1\dot\theta_2\cos(\theta_1-\theta_2)\right].\end{aligned}\]

This is the kinetic energy of the lower bob of a double pendulum with two arms of length \(a\) (Example 9.5), with the angles measured from the vertical through the point of suspension. Its cross term shows that \(\theta_1\) and \(\theta_2\), although orthogonal as variables, do not give an orthogonal metric.

9.2 Constraints and degrees of freedom

(a) Holonomic and non-holonomic constraints

A constraint is a restriction imposed on the motion of a system, such as an inextensible string, a rigid rod, or a surface on which a particle must remain. The elementary use of equations of constraint, to relate the accelerations of bodies connected by strings or sliding on wedges, was described in Chapter 1. Here we classify constraints by their mathematical form, since the form determines whether they can be used to reduce the number of coordinates.

A constraint is holonomic if it can be expressed as an equation connecting the coordinates of the particles and possibly the time,

\[f(\mathbf r_1,\mathbf r_2,\mathbf r_3,\dots,t)=0.\]

The rigid distance between two particles, \((\mathbf r_1-\mathbf r_2)^2-l^2=0\), and the condition that a particle remains on a given surface are holonomic. Each independent holonomic constraint can be used to eliminate one coordinate. The constraint equations are purely geometrical: they do not depend on the applied forces.

A constraint that cannot be written in this form is non-holonomic. One kind is expressed by an inequality, \(f(\mathbf r_1,\mathbf r_2,\dots,t)\geq0\). Examples are the molecules of a gas enclosed in a container, whose coordinates must lie within the walls, and a particle placed on the outside of a sphere of radius \(R\), for which \(r^2-R^2\geq0\); the particle stays on the surface only until it leaves it. Another kind is a relation between velocities that cannot be integrated into a relation between coordinates. A disc of radius \(r\) rolling along a straight line satisfies \(v=\omega r\), that is \(dx=r\,d\theta\), which integrates to \(x=r\theta+\text{const}\) and is therefore holonomic. A disc rolling upright on a plane, however, can change its heading, and its rolling conditions involve the orientation of the disc in a way that cannot be integrated; the constraint is non-holonomic and cannot be used to eliminate a dependent coordinate.

Constraints are classified independently by their time dependence. A constraint whose equation does not contain the time explicitly is scleronomic (or scleronomous), and one that does contain the time is rheonomic (or rheonomous). A bead on a fixed wire is subject to a scleronomic constraint; a bead on a wire that is made to rotate at a prescribed rate, or a pendulum whose point of support is moved in a prescribed manner, is subject to a rheonomic constraint.

(b) Degrees of freedom

The number of degrees of freedom of a system is the minimum number of independent coordinates needed to specify its configuration, that is, the number of independent motions it can perform. A system of \(N\) free particles has \(3N\) degrees of freedom. If the particles are subject to \(K\) independent holonomic constraints, the number of degrees of freedom is

\[\boxed{n=3N-K},\]

and this is the number of generalized coordinates required. A rigid body in three dimensions has six degrees of freedom, three for the position of a reference point and three for the orientation; in \(d\) dimensions it has \(d\) translational and \(\frac12d(d-1)\) rotational degrees of freedom.

For a simple pendulum of length \(a\) swinging in the \(xy\)-plane about the fixed point \(O\), the constraints are \(z=0\) and \(\sqrt{x^2+y^2}=a\), so \(n=3-2=1\). For a spherical pendulum of length \(a\) the only constraint is \(\sqrt{x^2+y^2+z^2}=a\), and \(n=3-1=2\).

Example 9.2 A particle of mass \(m\) is constrained to move under gravity on the inner surface of a cone of half-angle \(\alpha\) whose axis is vertical and whose vertex is at the origin (Figure 9.2). Write the equation of constraint (a) in Cartesian, (b) in cylindrical and (c) in spherical polar coordinates, and (d) find the number of degrees of freedom.

Figure 9.2 A particle on the inner surface of a cone of half-angle \(\alpha\).

Solution. (a) The line from the vertex to the particle always makes the angle \(\alpha\) with the vertical \(z\)-axis, so

\[\tan\alpha=\frac{\sqrt{x^2+y^2}}{z}.\]

With \(x=r\cos\theta\), \(y=r\sin\theta\) and \(z=z\), where \(r\) is now the distance from the axis, the constraint becomes \(\tan\alpha=r/z\), or \(r=z\tan\alpha\).
With \(x=r\sin\theta\cos\phi\), \(y=r\sin\theta\sin\phi\) and \(z=r\cos\theta\), \(\sqrt{x^2+y^2}/z=\tan\theta\), so the constraint is simply \(\theta=\alpha\). Spherical coordinates are therefore the natural choice: the constraint fixes one coordinate and leaves \(r\) and \(\phi\) free.
There is one particle and one constraint, so \(n=3\times1-1=2\). The two independent motions are along the slant generator of the cone (\(r\)) and round the axis (\(\phi\)).

Example 9.3 Two particles of masses \(m_1\) and \(m_2\) are connected by a string of length \(l\) that passes through a hole in a smooth horizontal table. The particle \(m_1\) moves on the table, and \(m_2\) hangs below the hole and moves along the vertical line through it (Figure 9.3). (a) Write down the equations of constraint. (b) Find the number of degrees of freedom.

Figure 9.3 Two particles connected by a string through a hole in a smooth table.

Solution. (a) Take the hole as origin, with the \(z\)-axis vertical. The particle \(m_1\) stays on the table, so \(z_1=0\). The particle \(m_2\) stays on the \(z\)-axis, so \(x_2=0\) and \(y_2=0\). The string has fixed length, so

\[\sqrt{x_1^2+y_1^2}+\sqrt{z_2^2}=l.\]

There are \(K=4\) constraints.

With \(N=2\), the number of degrees of freedom is \(n=3\times2-4=2\). A convenient pair of generalized coordinates is the distance \(r\) of \(m_1\) from the hole and its polar angle \(\theta\) on the table.

Example 9.4 The point of support \(P\) of a simple pendulum of length \(l\) and mass \(m\) moves on a vertical circle of radius \(a\) in the \(xy\)-plane, and the pendulum swings in the same plane (Figure 9.4). (a) Write down the equations of constraint. (b) Find the number of degrees of freedom, first when \(P\) is merely confined to the circle, and then when it is driven round the circle uniformly, at a prescribed angular frequency \(\gamma\).

Figure 9.4 A simple pendulum whose point of support \(P\) moves on a vertical circle.

Solution. (a) Take the centre \(O\) of the circle as origin, and let \(P\) be at \((x_1,y_1,z_1)\) and the bob at \((x_2,y_2,z_2)\). Both points move in the \(xy\)-plane, so \(z_1=0\) and \(z_2=0\). The point \(P\) stays on the circle, \(x_1^2+y_1^2=a^2\), and the length of the pendulum is fixed,

\[\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}=l.\]

The two moving points \(P\) and \(m\) count as \(N=2\) particles, although only one of them is massive. With the four constraints above, \(n=3\times2-4=2\): the angle of \(P\) on the circle and the angle \(\phi\) of the pendulum with the vertical. If, however, \(P\) is driven round uniformly, its angular position is prescribed, \(\theta_P=\gamma t\), which is a fifth, rheonomic, constraint. Then \(K=5\) and \(n=1\), the single coordinate being \(\phi\). This case is treated in Example 9.20.

Example 9.5 A double pendulum consists of a particle \(m_1\) suspended from a fixed point \(O\) by a string of length \(l_1\), and a particle \(m_2\) suspended from \(m_1\) by a string of length \(l_2\), both moving in the same vertical \(xy\)-plane (Figure 9.5). Write down the equations of constraint, find the number of degrees of freedom, and obtain the kinetic and potential energies.

Figure 9.5 A double pendulum in the vertical \(xy\)-plane.

Solution. The particle \(m_1\) moves in the \(xy\)-plane at fixed distance from \(O\): \(z_1=0\) and \(\sqrt{x_1^2+y_1^2}=l_1\). The particle \(m_2\) moves in the same plane at fixed distance from \(m_1\): \(z_2=0\) and \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}=l_2\). With \(N=2\) and \(K=4\) there are \(n=2\) degrees of freedom, and the angles \(\theta_1\) and \(\theta_2\) of the two strings with the downward vertical serve as generalized coordinates. Then

\[\begin{aligned}x_1&=l_1\sin\theta_1,&y_1&=-l_1\cos\theta_1,\\x_2&=l_1\sin\theta_1+l_2\sin\theta_2,&y_2&=-l_1\cos\theta_1-l_2\cos\theta_2.\end{aligned}\]

The velocity of \(m_2\) is the sum of two vectors of magnitudes \(l_1\dot\theta_1\) and \(l_2\dot\theta_2\) inclined at the angle \(\theta_1-\theta_2\), exactly as in Example 9.1. Hence

\[T=\frac12(m_1+m_2)l_1^2\dot\theta_1^2+\frac12m_2l_2^2\dot\theta_2^2+m_2l_1l_2\dot\theta_1\dot\theta_2\cos(\theta_1-\theta_2),\]

\[V=m_1gy_1+m_2gy_2=-(m_1+m_2)gl_1\cos\theta_1-m_2gl_2\cos\theta_2.\]

9.3 D'Alembert's principle and Lagrange's equations

(a) Virtual displacements and virtual work

A virtual displacement \(\delta\mathbf r_i\) of a system is an arbitrary infinitesimal change of the coordinates of its particles, consistent with the constraints as they stand at a given instant \(t\). The displacement is called virtual to distinguish it from an actual displacement \(d\mathbf r_i\) of the system in a time \(dt\), during which the forces and constraints may change. Time is held fixed in a virtual displacement.

Suppose first that the system is in equilibrium, so that the total force on each particle vanishes, \(\mathbf F_i=0\). Then the virtual work \(\mathbf F_i\cdot\delta\mathbf r_i\) vanishes for each particle, and so does the sum, \(\sum_i\mathbf F_i\cdot\delta\mathbf r_i=0\). We now separate the force on each particle into the applied force \(\mathbf F_i^{(a)}\) and the force of constraint \(\mathbf f_i\), so that

\[\sum_i\mathbf F_i^{(a)}\cdot\delta\mathbf r_i+\sum_i\mathbf f_i\cdot\delta\mathbf r_i=0.\]

We restrict ourselves to systems for which the net virtual work of the forces of constraint is zero. This holds for a rigid body, whose internal forces act in equal and opposite pairs along the lines joining the particles, and for a large number of other constraints. A particle constrained to move on a smooth surface is acted upon by a normal force of constraint, while the virtual displacement is tangent to the surface, so the virtual work vanishes. This is true even when the surface moves with time: the constraint force is then still perpendicular to the surface at each instant, so its virtual work is zero, although its work in an actual displacement during \(dt\) need not vanish. The condition also holds for rolling without slipping, since the friction force acts at a point that is momentarily at rest. It fails when sliding friction is present, and such systems are excluded from the formulation. With this restriction the condition of equilibrium becomes

\[\boxed{\sum_i\mathbf F_i^{(a)}\cdot\delta\mathbf r_i=0},\]

the principle of virtual work. The coefficients of \(\delta\mathbf r_i\) cannot be set equal to zero separately, that is, in general \(\mathbf F_i^{(a)}\neq0\), because the \(\delta\mathbf r_i\) are not independent but are connected by the constraints. To make use of the principle we must express it in terms of independent virtual displacements \(\delta q_j\) of the generalized coordinates.

(b) D’Alembert’s principle

The principle of virtual work deals only with statics. To obtain a condition that covers the general motion of a system we use a device due to James Bernoulli and developed by D’Alembert. The equation of motion \(\mathbf F_i=\dot{\mathbf p}_i\) can be written \(\mathbf F_i-\dot{\mathbf p}_i=0\), which states that the particles would be in equilibrium under the actual forces together with a “reversed effective force” \(-\dot{\mathbf p}_i\). Hence \(\sum_i(\mathbf F_i-\dot{\mathbf p}_i)\cdot\delta\mathbf r_i=0\), and making the same separation into applied forces and forces of constraint, and again assuming that the constraint forces do no virtual work, we obtain

\[\boxed{\sum_i\left(\mathbf F_i^{(a)}-\dot{\mathbf p}_i\right)\cdot\delta\mathbf r_i=0},\]

which is D’Alembert’s principle. The forces of constraint no longer appear, and the superscript \((a)\) may now be dropped without ambiguity.

(c) Lagrange’s equations

Let the system have \(n\) degrees of freedom and holonomic constraints, so that the position vectors can be written in terms of \(n\) independent generalized coordinates, \(\mathbf r_i=\mathbf r_i(q_1,\dots,q_n,t)\). Then

\[\mathbf v_i=\dot{\mathbf r}_i=\sum_j\frac{\partial\mathbf r_i}{\partial q_j}\dot q_j+\frac{\partial\mathbf r_i}{\partial t},\qquad \delta\mathbf r_i=\sum_j\frac{\partial\mathbf r_i}{\partial q_j}\delta q_j,\]

the second relation containing no term in \(\delta t\) because time is fixed in a virtual displacement. Two identities follow from the first relation. Since \(\mathbf v_i\) is linear in the \(\dot q_j\) with coefficients \(\partial\mathbf r_i/\partial q_j\),

\[\frac{\partial\mathbf v_i}{\partial\dot q_j}=\frac{\partial\mathbf r_i}{\partial q_j},\]

and since the order of differentiation with respect to \(t\) and \(q_j\) may be interchanged,

\[\frac{d}{dt}\left(\frac{\partial\mathbf r_i}{\partial q_j}\right)=\sum_k\frac{\partial^2\mathbf r_i}{\partial q_k\partial q_j}\dot q_k+\frac{\partial^2\mathbf r_i}{\partial t\,\partial q_j}=\frac{\partial\mathbf v_i}{\partial q_j}.\]

The virtual work of the applied forces becomes

\[\sum_i\mathbf F_i\cdot\delta\mathbf r_i=\sum_jQ_j\,\delta q_j,\qquad\text{where}\qquad \boxed{Q_j=\sum_i\mathbf F_i\cdot\frac{\partial\mathbf r_i}{\partial q_j}}\]

is the generalized force associated with \(q_j\). The product \(Q_j\,\delta q_j\) always has the dimensions of work, so \(Q_j\) is a force when \(q_j\) is a length and a torque when \(q_j\) is an angle. The other term of D’Alembert’s principle is

\[\sum_i\dot{\mathbf p}_i\cdot\delta\mathbf r_i=\sum_j\sum_im_i\ddot{\mathbf r}_i\cdot\frac{\partial\mathbf r_i}{\partial q_j}\,\delta q_j.\]

Using the two identities,

\[m_i\ddot{\mathbf r}_i\cdot\frac{\partial\mathbf r_i}{\partial q_j}=\frac{d}{dt}\left(m_i\mathbf v_i\cdot\frac{\partial\mathbf r_i}{\partial q_j}\right)-m_i\mathbf v_i\cdot\frac{d}{dt}\frac{\partial\mathbf r_i}{\partial q_j}=\frac{d}{dt}\left(m_i\mathbf v_i\cdot\frac{\partial\mathbf v_i}{\partial\dot q_j}\right)-m_i\mathbf v_i\cdot\frac{\partial\mathbf v_i}{\partial q_j}.\]

Summing over \(i\) and introducing the kinetic energy \(T=\sum_i\frac12m_iv_i^2\), this is \(\dfrac{d}{dt}\dfrac{\partial T}{\partial\dot q_j}-\dfrac{\partial T}{\partial q_j}\). D’Alembert’s principle therefore takes the form

\[\sum_j\left[\frac{d}{dt}\left(\frac{\partial T}{\partial\dot q_j}\right)-\frac{\partial T}{\partial q_j}-Q_j\right]\delta q_j=0.\]

For holonomic constraints the \(\delta q_j\) are independent, so each coefficient vanishes separately:

\[\boxed{\frac{d}{dt}\left(\frac{\partial T}{\partial\dot q_j}\right)-\frac{\partial T}{\partial q_j}=Q_j},\qquad j=1,\dots,n.\]

If the applied forces are derivable from a potential energy \(V(q,t)\) that does not depend on the velocities, then \(Q_j=-\partial V/\partial q_j\) and \(\partial V/\partial\dot q_j=0\). Defining the Lagrangian

\[\boxed{L(q_j,\dot q_j,t)=T-V},\]

the equations become

\[\boxed{\frac{d}{dt}\left(\frac{\partial L}{\partial\dot q_j}\right)-\frac{\partial L}{\partial q_j}=0},\qquad j=1,\dots,n.\]

These are Lagrange’s equations of motion. They are \(n\) second-order differential equations, one for each degree of freedom, and they contain neither the forces of constraint nor any reference to a particular coordinate system: their form is the same whatever generalized coordinates are chosen. The same equations follow from a variational principle, Hamilton’s principle, which we develop after a short account of the calculus of variations.

9.4 The calculus of variations

(a) The Euler–Lagrange equation

A functional is a quantity that depends on a whole function rather than on a finite number of variables. Consider the definite integral

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Chapter 9 Lagrangian Formulation