3
Analytical dynamics—an
overview
3.1 Introduction
In order to describe the motion of a physical system, it is necessary to specify
its position in space and time. Strictly speaking, only relative motion is
meaningful, because it is always implied that the description is made with
respect to some observer or frame of reference.
In accordance with the knowledge of his time, Newton regarded the
concepts of length and time interval as absolute, which is to say that these
quantities are the same in all frames of reference. Modern physics showed
that Newton’s assumption is only an approximation but, nevertheless, an
excellent one for most practical purposes. In fact, Newtonian mechanics,
vastly supported by experimental evidence, is the key to the explanation of
the great majority of everyday facts involving force and motion.
If one introduces as a fundamental entity of mechanics the convenient
concept of material particle—that is, a body whose position is completely
defined by three Cartesian coordinates x, y, z and whose dimension can be
neglected in the description of its motion—Newton’s second law reads
(3.1)
where F is the resultant (i.e. the vector sum) of all the forces applied to the
particle,
is the particle acceleration and the quantity m
characterizes the material particle and is called its mass. Obviously, x is here
the vector of components x, y, z.
Equation (3.1) must not be regarded as a simple identity, because it establishes
a form of interaction between bodies and thereby describes a law of nature; this
interaction is expressed in the form of a differential equation that includes only
the second derivatives of the coordinates with respect to time. However, eq
(3.1) makes no sense if the frame of reference to which it is referred is not
specified. A difficulty then arises in stating the cause of acceleration: it may be
either the interaction with other bodies or it may be due to some distinctive
properties of the reference frame itself. Taking a step further, we can consider a
set of material particles and suppose that a frame of reference exists such that
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all accelerations of the particles are a result of their mutual interaction. This can
be verified if the forces satisfy Newton’s third law, that is they are equal in
magnitude and opposite in sign for any given pair of particles.
Such a frame of reference is called inertial. With respect to an inertial
frame of reference a free particle moves uniformly in a straight line and
every observer in uniform rectilinear motion with respect to an inertial frame
of reference is an inertial observer himself.
3.2 Systems of material particles
Let us consider a system of N material particles and an inertial frame of
reference. Each particle is subjected to forces that can be classified either as:
internal forces, due to the other particles of the system or external forces,
due to causes that are external to the system itself. We can write eq (3.1) for
the kth particle as
(3.2)
where k=1, 2,…, N is the index of particle,
and
are the resultants
of external and internal forces, respectively. In addition, we can write the
resultant of internal forces as
(3.3)
which is the vector sum of the forces due to all the other particles, Fkj being the
force on the kth particle due to the jth particle. Newton’s third law states that
(3.4)
hence
(3.5)
and eq (3.2), summing on the particle index k, leads to
(3.6)
These results are surely well known to the reader but, nevertheless, they
are worth mentioning because they show the possibility of writing equations
where internal forces do not appear.
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3.2.1 Constrained systems
Proceeding further in our discussion, we must account for the fact that, in
many circumstances, the particles of a given system are not free to occupy
any arbitrary position in space, the only limitation being their mutual
influences. In other words, we must consider constrained systems, where the
positions and/or the velocities of the particles are connected by a certain
number of relationships that limit their motion and express mathematically
the equations of constraints.
A perfectly rigid body is the simplest example: the distance between any
two points remains unchanged during the motion and 3N–6 equations (if
and the points are not aligned) must be written to satisfy this condition.
In every case, a constraint implies the presence of a force which may be, a
priori, undetermined both in magnitude and direction; these forces are called
reaction forces and must be considered together with all other forces. For the
former, however, a precise law for their dependence on time, coordinates or
velocities (of the point on which they act or of other points) is not given; when
we want to determine the motion or the equilibrium of a given system, the
information about them is supplied by the constraints equations.
Constraints, in turn, may be classified in many ways according to their
characteristics and to the mathematical form of the equations expressing
them, we give the following definitions: if the derivatives of the coordinates
do not appear in a constraint equation we speak of holonomic constraint
(with the further subdivision in rheonomic and scleronomic), their general
mathematical expression being of the type
(3.7)
where time t appears explicitly for a rheonomic constraint and does not
appear for a scleronomic one. In all other cases, the term nonholonomic is
used.
For example, two points rigidly connected at a distance L must satisfy
(3.8)
A point moving in circle in the x–y plane must satisfy
(3.9a)
or, in parametric form,
(3.9b)
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where the angle θ (the usual angle of polar coordinates) is the parameter
and shows how this system has only one degree of freedom (the angle θ), a
concept that will be defined soon.
Equations (3.8) and (3.9) are two typical examples of holonomic
(scleronomic) constraints. A point moving on a sphere whose radius increases
linearly with time (R=at) is an example of rheonomic constraint, the constraint
equation being now
(3.10a)
or, in parametric form
(3.10b)
where the usual angles for spherical coordinates have been used.
From the examples above, it can be seen that holonomic constraints reduce
the number of independent coordinates necessary to describe the motion of
a system to a number that defines the degrees of freedom of the system.
Thus, a perfectly rigid body of N material points can be described by only
six independent coordinates; in fact, a total of 3N coordinates identify the
N points in space but the constraints of rigidity are expressed by 3N–6
equations, leaving only six degrees of freedom.
Consider now a sphere of radius R that rolls without sliding on the x–y
plane. Let X and Y be the coordinates of the centre with respect to the fixed
axes x and y and let ωx and ωy be the components of the sphere angular
velocity along the same axes. The constraint of rolling without sliding is
expressed by the equations (the dot indicates the time derivative)
(3.11a)
which are the projections on the x and y axes of the vector equation
where v and ω are the sphere linear and angular velocities, the symbol × indicates
the vector product, C is the position vector of the point of contact of the sphere
with the x–y plane and O is the position vector of the centre of the sphere. Making
use of the Euler angles φ, θ and
(e.g. Goldstein [1]), eqs (3.11a) become
(3.11b)
which express the nonholonomic constraint of zero velocity of the point C.
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A disc rolling without sliding on a plane is another classical example—
which can be found on most books of mechanics—of nonholonomic
constraint. These constraints do not reduce the number of degrees of freedom
but only limit the way in which a system can move in order to go from one
given position to another.
In essence, they are constraints of ‘pure mobility’: they do not restrict the
possible configurations of the system, but how the system can reach them.
Obviously, a holonomic constraint of the kind (dropping the vector
notation and using scalar quantities for simplicity)
implies the relation on the time derivative
but when we have a general relation of the kind
which cannot be obtained by differentiation (i.e. it is not an exact differential)
the constraint is nonholonomic. In the equation above, A and B are two
functions of the variables x1 and x2. Incidentally, it may be interesting to
note that the assumption of relativistic mechanics stating that the velocity of
light in vacuum
is an upper limit for the velocities of
physical bodies is, as a matter of fact, a good example of nonholonomic
constraint.
Thus, in the presence of constraints:
1. The coordinates xk are no longer independent (being connected by the
constraint equations).
2. The reaction forces appear as unknowns of the problem; they can only
be determined a posteriori, that is, they are part of the solution itself.
This ‘indetermination’ is somehow the predictable result of the fact that
we omit a microscopical description of the molecular interactions
involved in the problem and we make up for this lack of knowledge
with information on the behaviour of constraints—the reaction forces—
on a macroscopic scale. So, unless we are specifically interested in the
determination of reaction forces, it is evident the interest in writing, if
possible, a set of equations where the reaction forces do not appear.
For every particle of our system we must now write
(3.12)
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where F is the resultant of active (internal and external) forces, Φ is the
resultant of reactive (internal and external) forces and the incomplete
knowledge on Φ is supplied by the equation(s) of constraint.
The nature of holonomic constraints itself allows us to tackle point (1) by
introducing a set of generalized independent coordinates; in addition, we
are led to equations where reaction forces disappear if we restrict our interest
to reactive forces that, under certain circumstances of motion, do no work.
These are the subjects of the next section.
3.3 Generalized coordinates, virtual work and d’Alembert
principles: Lagrange’s equations
If m holonomic constraints exist between the 3N coordinates of a system of
N material particles, the number of degrees of freedom is reduced to
n=3N–m. It is then possible to describe the system by means of n configuration
parameters
usually called generalized coordinates, which are
related to the Cartesian coordinates by a transformation of the form
(3.13)
and time t does not appear explicitly if the constraints are not time dependent.
The advantage lies obviously in the possibility of choosing a convenient set
of generalized coordinates for the particular problem at hand.
From eq (3.13) we note that the velocity of the kth particle is given by
(3.14)
Let us now define the kth (k=1, 2,…, N) virtual displacement δxk as an
infinitesimal displacement of the kth particle compatible with the constraints.
In performing this displacement we assume both active and reactive forces
to be ‘frozen’ in time at the instant t, that is to say that they do not change
as the system passes through this infinitesimal change of its configuration.
This justifies the term ‘virtual’, as opposed to a ‘real’ displacement dxk, which
occurs in a time dt. Similarly, we can define the kth virtual work done by
active and reactive forces as
and the total work on all of the N particles is
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(3.15)
If the system is in equilibrium, eq (3.15) is zero, because each one of the N
terms in parentheses is zero; if in addition we restrict our considerations to
reactive forces whose virtual work is zero, eq (3.15) becomes
(3.16)
which expresses the principle of virtual work. We point out that only active
forces appear in eq (3.16) and, in general,
because the virtual
displacements are not all independent, being connected by the constraints
equations. The method leads to a number of equations
holonomic constraints—to the number of degrees of freedom.
The assumption on the constraints leading to eq (3.16) is not very restrictive
and, in practice, is valid for all holonomic constraints without friction. When
the constraints are not frictionless, the equation is still valid if we count the
tangential components of friction forces as active forces themselves. It could
be added that if the principle of virtual work allows to obtain the equilibrium
condition with frictionless constraints, the same conditions must apply when
friction is present.
Since holonomic constraints are our major concern, we consider the
transformation (3.13) and write
(3.17)
where time does not appear because of the definition of virtual displacement.
Substitution in eq (3.16) gives the principle of virtual work
(3.18)
in terms of the generalized forces, defined as
(3.19)
The generalized forces do not necessarily have the dimensions of a force
themselves, but the product Qjδqj has the dimension of work. Now all the
δqjs are independent and eq (3.18) implies
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(3.20)
for every j=1, 2,…, n. Equation (3.20) expresses the equilibrium condition
of our system and, as such, applies to the static case.
The extension to the dynamic case—with the difference that now we want
to determine the equations of motion—is made by means of d’Alembert’s
principle, which sees eq (3.1) as an equilibrium equation by writing
(3.21)
which means that a material particle is in equilibrium if we consider the
inertia force (–ma) together with all other forces whose resultant is F. Under
the same assumptions that lead to eq (3.16), we can rewrite it in the form
(3.22)
and transform it into an equation where the virtual displacements of the
generalized coordinates appear, so that we are allowed to say that every
multiplicative coefficient of these virtual displacements is individually equal
to zero.
We have already considered the term
for the other term we
can substitute eq (3.17) into eq (3.22) and obtain
(3.23)
For the summation on the index k we can write
(3.24)
and since it is not difficult to verify that (eq (3.14))
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eq (3.24) becomes
Substituting back in eq (3.23), we get the expression
where we recognize the total kinetic energy of the system
(3.25)
which, in general, is a function of the kind
or,
for short,
Putting all the pieces back together we finally get the expression we were
looking for, i.e.
from which it follows that
(3.26)
owing to the independence of the δqjs. Equations (3.26) are called Lagrange’s
equations of the second kind and are valid for holonomic systems with
frictionless constraints. It can be shown that they form a system of n secondorder differential equations which can be solved for the second derivatives
and written as
(3.27)
The solution is determined completely by introducing 2n constants of
integration, obtained by imposing the initial conditions at t=0.
When the system is conservative (see next section for more details), the
as
forces can be obtained from a scalar function)
(3.28)
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The elementary work, i.e. the work done as the system passes through an
infinitesimal displacement, is an exact differential and eq (3.18) can be
written as
which implies
(3.29)
Lagrange’s equations (3.26) thus become
and defining the Lagrangian function (or simply Lagrangian) as the difference
between kinetic and potential energies, i.e.
(3.30)
we obtain Lagrange’s equations of the first kind for conservative systems as
(3.31)
where
and we exploited the fact that
When some of the forces acting on the system are conservative and some
others are not, it is worth noting that Lagrange’s equations can always be
written in the form of eq (3.31), where now a term Qj appears on the righthand side. In this case, the potential V accounts for conservative forces and
the Qjs represent all the forces that cannot be derived from a potential function.
A fundamental property of Lagrange’s equations (3.26) and (3.31) is that
they are invariant under an arbitrary transformation of generalized coordinates;
in fact, it can be proven that an invertible transformation of the type
(3.32)
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converts, for example, eqs (3.31) into
(3.33)
where L1 is the appropriate Lagrangian in the new coordinates.
3.3.1 Conservative forces
It is well known that the work done by a force F on a material particle
which undergoes an infinitesimal displacement dx is
(3.34)
Suppose now that the particle moves along some curve in space (say,
from a point A to a point B), with the force varying as the particle moves.
On a curve, x, y and z are related by the equations of the curve and in three
dimensions two equations are needed. Thus, along a curve there is only one
independent variable and the total work from A to B, i.e. the integral of eq
(3.34) is an ordinary integral of a function of one variable, more precisely,
it is a line integral. To evaluate a line integral, we must write it as a single
integral using one independent variable.
For example, in two dimensions (the x–y plane), given the force field
where i and j are, respectively, the usual unit vectors in the positive x and y
directions, let us find the work from A=(0, 0) to B=(2, 1) along the two
paths:
1. straight line
2. parabola
It is not difficult to see that
becomes an integral in dx and leads to the following results: W=1 in case
(1), and W=2/3 in case (2). So, the work done may depend on the path the
particle follows; in fact, it usually will when there is friction.
A force field for which the quantity
(3.35)
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depends upon the path as well as the endpoints is called nonconservative.
Physically this means that energy has been dissipated, for example by friction.
However, there are conservative fields for which the integral above is the
same between two given points, regardless of what path we calculate it along.
It can be shown from calculus that, ordinarily,
(3.36)
is a necessary and sufficient condition for the integral (3.35) to be independent
of path, that is
for conservative fields and
for
nonconservative fields. Explicitly, the components of the vector
can
be obtained from the determinant
It is not difficult to justify the considerations above. Suppose that for a
given F there is a function W(x, y, z) such that
then, from the fact that
etc., we see that the
components of
are all zero. Then, if
it follows that
Conversely, if
then we can find a function W(x, y, z) for which
In this case we can write
(3.37)
and
(3.38)
where W(B) and W(A) are the values of the function W at the endpoints of
the path of integration. Since the integral does not depend on the path but
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only on the endpoints A and B, then F is conservative, W is the scalar
potential of the force F and it is customary to define the potential energy
as V=–W.
From its definition, it is clear that V can be changed by adding any
constant; this corresponds to the choice of the zero level of the potential
energy and has no effect on F.
The differential dW of eq (3.37) is an exact differential. In the light of the
discussion above, we can say that
is a necessary and sufficient
condition for F . dx to be an exact differential. More generally, the following
theorem of vector calculus can be proven:
If the components of F have continuous first partial derivatives in a simply
connected region, then any one of the following five conditions implies all
the others:
1.
at every point of the region.
2.
around every simple closed curve in the region.
3. F is conservative, that is,
does not depend on the path of
integration from point A to point B (the path, obviously, must lie entirely
in the region).
4. F · dx is an exact differential.
5. There exists a function V such that
where V is single-valued.
Generally speaking, a region is ‘simply connected’ if any simple closed curve
in the region can be shrunk to a point without encountering any points not
in the region.
Two examples of conservative forces that will be of interest to us are
gravitational forces and elastic forces. It is very well known to the reader
that the potential energy of a body of mass m lifted above the surface of the
earth to a height h is mgh (where h=0 is the choice for zero potential energy)
and the potential energy of a stretched (or compressed) spring within its
linear range is
In fact,
where ∆l is the
displacement from the unstretched position, which is commonly assumed as
the position of zero potential energy.
The elastic potential energy is also called strain energy; in general, for an
elastically deformed body it can be written as
where wstrain is the strain energy per unit volume of the body, dV is the
element of volume and we have used here the letter E for the energy to
avoid ambiguities between the usual notation V for the potential energy
and V for volume. Again, zero potential energy is assigned to the undeformed
body.
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Example 3.1. Illustratively, we can calculate the potential energy of a rod of
length L and cross-section A under the action of an axial tensile force F(x, t).
The infinitesimal element dx of the rod undergoes an elongation
where u(x, t) is the displacement and ε(x, t) is the strain at point x and time
t. The strain energy of the volume element Adx is equal in magnitude to the
work done by the force F, i.e.
from the definition of axial stress
within the elastic range—so that
and the assumption to remain
(E=Young’s modulus)—we get
from which the strain energy of the rod follows as
(3.39)
Obviously, the action of an axial compressive force F(x, t) leads to the same
result.
3.3.2
A few generalizations of Lagrange’s equations
One of the most important nonconservative forces is viscous damping.
The viscous force is proportional to the particle velocity, so that for the
kth particle
(3.40a)
or, explicitly, for the single components
(3.40b)
where k is the particle index (k=1, 2,…, N) and α is the component index
(indicates the x component,
they component and
the z component).
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These forces can be derived from a scalar function D—a kind of
generalized potential—of the form
(3.40c)
from which it follows that
or
(3.41)
where the subscript vk indicates that the gradient is taken with respect to
velocities. For example, in the case of a single particle moving in one direction
with velocity v the equations above simply state that F=–cv and D=cv2/2, so
that F=–dD/dv.
From eq (3.41) we get
and since
it follows that
(3.42)
Lagrange’s equation may then be written as
(3.43)
in the case of conservative and viscous forces, or as
(3.44)
when conservative, viscous and nonconservative (other than viscous) forces
are acting on our system. The first two terms on the left-hand side account
for conservative forces (through the function V which is part of the
Lagrangian L), the third term accounts for viscous forces and the term on
the right-hand side accounts for all other nonconservative forces.
Sometimes it may be convenient to work with constrained coordinates. Let us
suppose that we have an n-degrees of freedom system and n+1 coordinates
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(which we will call)
), i.e. we have one coordinate in excess with
respect to the minimum number required to describe our system. Therefore,
they will be connected by one constraint equation of the general form
(3.45)
from which it follows that
(3.46)
and implies that only n out of the n+1 δ qis are independent, the (n+1)th
being determined by eq (3.46). The application of d’Alembert’s principle to
the principle of virtual work leads again to the expression
(3.47)
but now this does not imply that the coefficients in brackets are individually
zero. Let us then add times eq (3.46) to eq (3.47), where is an unknown
arbitrary parameter called the Lagrangian multiplier,
(3.48)
and let us choose
so that
(3.49)
The sum (3.48) is then a sum of n terms implying the n Lagrange equations
which, together with eqs (3.49) and (3.45), are n+2 equations in the n+2
unknowns
The extension to the case of more than one,
say N (N>n), coordinates is straightforward; we will then have m=N–n
constraint equations and m Lagrangian multipliers
thus
obtaining N+m equations in N+m unknowns, i.e. the N coordinates plus
the m multipliers.
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The reader has probably noticed that in the foregoing discussion on
Lagrangian multipliers we assumed one (or more) holonomic constraint (eq
(3.45)). The method can be used also in the case of nonholonomic constraints
which, very often, have the general form
(3.50)
where l=1, 2,…, m in the case of m nonholonomic constraints. The generalized
coordinates are now n in number,
and bl=bl(t). In this case
it is not possible to find m functions such that eqs (3.50) are equivalent to
fl=const, but referring to virtual displacements we can write eq (3.50) in
differential form
multiply it by l , sum it to the usual expression (3.47), where now j=1, 2,…, n,
choose the m multipliers so that, say, the last m terms are zero, i.e.
(3.51)
and obtain explicitly the Lagrange multipliers; it follows that
(3.52)
Equations (3.50) and (3.52), are now the n equations that can be solved
for the n coordinates qj. Equivalently, we can say that eqs (3.50), (3.51) and
(3.52) are n+m equations in n+m unknowns.
The method is particularly useful in problems of statics because the
parameters
represent the generalized reactive forces which
automatically appear as a part of the solution itself.
3.3.3 Kinetic energy and Rayleigh’s dissipation function in
generalized coordinates. Energy conservation
From the foregoing discussion it is clear that Lagrange’s equations are based
on the calculation of the derivatives of the kinetic energy with respect to the
generalized coordinates. Since we know that
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from eq (3.14) we get the kinetic energy as the sum of three terms
(3.53)
where
(3.53a)
(3.53b)
(3.53c)
and a noteworthy simplification occurs in the case of scleronomic (time
independent) constraints, i.e. when time does not appear explicitly in the
coordinate transformation (3.13). In fact, in this case, only the third term in
eq (3.53) is different from zero and the kinetic energy is a quadratic form in
the generalized velocities.
First we note that
(3.54)
and the term in question can also be written in matrix notation as
(3.55)
where
A is symmetrical and
is simply the transpose of matrix
Furthermore, by virtue of Euler’s theorem on homogeneous functions—
which we state briefly without proof—T is a homogeneous function of
degree 2 in the s and can be written as
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(3.56)
a form that may turn out to be useful in many circumstances.
Euler’s theorem on homogeneous functions. A general function f(x, y, z) is
homogeneous of degree m if
Euler’s theorem says that if f is homogeneous of degree m then
By looking at the general expression of Rayleigh’s dissipation function
(eq (3.40c)) it is not difficult to see that this function too is a quadratic form
in the generalized velocities which can be written as
(3.57)
where
(3.57a)
and
(3.57b)
From eq (3.56) it follows that
(3.58)
On the other hand, since
we can also write
(3.59)
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subtracting eq (3.59) from (3.58) and using Lagrange’s equations we get
(3.60)
If the forces are conservative, i.e.
side is nothing but –dV/dt so that
the term on the right-hand
(3.61)
or
(3.62)
which states the conservation of energy for a conservative system with
scleronomic constraints.
If the nonconservative forces are viscous in nature, the term on the righthand side of eq (3.60) is the power dissipated by such forces; following the
same line of reasoning that leads to eq (3.56) we have
hence
(3.63)
and, as expected, energy is not conserved in this case, 2D being the rate of
energy dissipation due to the frictional forces.
3.3.4 Hamilton’s equations
Lagrange’s equations for a n-degree-of-freedom system are n second-order
equations. In many cases, such a set of equations is equivalent to a set of 2n
first order equations; in analytical mechanics these are Hamilton’s equations,
which we introduce briefly. Whenever a Lagrangian function exists, we can
define the conjugate momenta as
(3.64)
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and a Hamiltonian function
(3.65)
where the
are expressed as functions of the generalized momenta and
then H is, in general, a function
By differentiating eq (3.65)
and taking into account Lagrange’s equations, we get
which is valid for any choice of the differentials dpj and dq j; it then follows
that
(3.66)
These are Hamilton’s canonical equations, showing a particular formal
elegance and symmetry. It is not difficult to show that in the case of
scleronomic constraints the function H is the total energy T+V of the system,
which is a constant of motion when the Lagrangian does not depend explicitly
on time t. Since we will make little use of Hamilton’s equations, we do not
pursue the subject any further and the interested reader can find references
for additional reading at the end of Part I.
3.4
Hamilton’s principle of least action
Many problems of physics and engineering lend themselves to mathematical
formulations that belong to the specific subject called ‘calculus of variations’.
In principle, the basic problem is the same as finding the maximum and
minimum values of a function f(x) in ordinary calculus. We calculate df/dx
and set it equal to zero; the values of x that we find are called stationary
points and they correspond to maximum points, minimum points or points
of inflection with horizontal tangent. In the calculus of variation the quantity
to make stationary is an integral of the form
(3.67)
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where
The statement of the problem is as follows: given the two
points
and the form of the function f, find the curve
y=y(x), passing through the given points, which makes the integral I have the
smallest value (or stationary value). Such a function y(x) is called an extremal.
Then let η(x) be an arbitrary function whose only properties are to be zero at
x1 and x2 and have continuous second derivatives. We define Y(x) as
(3.68)
where y(x) is the desired (unknown) extremal and e is a parameter. Due to
the arbitrariness of η(x), Y(x) represents any curve that can be drawn through
the points
. Then
and we want
when
. By differentiating under the integral sign
and substituting eq (3.68) we are led to
(3.69)
because Y=y when
. The second term can be integrated by parts as
and the first term on the right-hand side is zero because of the requirements
on η(x). We obtain
and for the arbitrariness of η(x)
(3.70)
which is called the Euler-Lagrange equation. The extension to more than
one dependent variable (but always one independent variable) is
straightforward: one gets as many Euler-Lagrange equations as the number
of dependent variables.
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A word of warning about the mathematical symbols: the reader will often
find δI for the differential
where the δ symbol reads ‘the
variation of’ and indicates that ε and not x is the differentiation variable.
Similarly,
and
The formal structure of eq (3.70) already suggests the connection with
the problems of interest to us. In fact, if we replace the independent variable
x with time and if the function f is the Lagrangian of some mechanical system,
eq (3.70) is a Lagrange equation for a single-degree-of-freedom conservative
system where y(x)—the dependent variable—is nothing but the generalized
coordinate q(t).
In the case of n degrees of freedom we have n generalized coordinates
and the procedure outlined above leads exactly to the n Lagrange equations
(3.31). The integral to be made stationary is called action; it is usually denoted
by the symbol S and is defined as
(3.71)
where L is the Lagrangian of the system under consideration. Hamilton’s
principle of least action can then be written
(3.72)
Thus, for a conservative system with frictionless constraints the natural
motion has the property of making the Hamiltonian action stationary with
respect to all varied motions for which
More specifically, the class of varied motions considered in this case are
called ‘synchronous’ to make the increments δq correspond to the virtual
displacements introduced in preceding sections; other types of increments
can be considered but this is beyond the scope of the present discussion.
In essence, Hamilton’s principle is an integral version of the virtual work
principle because it considers the entire motion of the system between two
instants t1 and t2. Furthermore, Hamilton’s principle is a necessary and
sufficient condition for the validity of Lagrange’s equations; as such, it can
be obtained by starting from d’Alembert’s principle, i.e. the equation
(e.g. Meirovitch [2]). The advantages—as with Lagrange’s and Hamilton’s
equations—are the possibility of working with a scalar function, the
invariance with respect to the coordinate system used and their validity for
linear and nonlinear systems.
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An extended, or generalized Hamilton principle can be stated as
(3.73)
where
is the virtual work done by nonconservative forces (including damping forces
and forces not accounted for in V).
Example 3.2. As an example, we can use the generalized Hamilton principle
to derive the governing equation of motion for the transverse vibration of
an Euler-Bernoulli beam (shear deformation and rotary inertia ignored, see
Chapter 8 for more details) in the general case of variable mass density and
bending stiffness EI(x), as shown in Fig. 3.1.
The kinetic energy is
(3.74)
where
is the mass per unit length and y=y(x, t) is the transverse
displacement from the neutral axis; the potential strain energy is in this case
(3.75)
where E is Young’s modulus and I(x) is the cross-sectional area moment of
inertia of the beam about its neutral axis. Finally, the virtual work done by the
Fig. 3.1 Transverse vibration of a slender (Euler-Bernoulli) beam.
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nonconservative distributed external force f(x, t) is
(3.76)
where
is an arbitrary virtual displacement.
Inserting the expressions above in the generalized Hamilton principle,
under the assumptions that the operator δ commutes with the time and spatial
derivatives and that the time and spatial integrations are interchangeable,
we obtain, for example, for the term
Integration by parts with respect to time gives
(3.77)
because δy vanishes at t=t1 and t=t2. Similar calculations apply for the strain
energy term, where now two integrations by part with respect to x must be
performed and we get
Putting all the pieces back together in Hamilton’s principle we obtain
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