1.5.2 Hamilton’s Principle
Hamilton’s principle holds for any mechanical systems subjected to monogenic
forces and holonomic constraints (to be explained). By contrast to Lagrange’s
equations, it applies too for systems characterized by infinitely many degrees of
freedom, that is, for continuous or distributed systems. By Hamilton’s principle,
dH ¼ 0; where H
Z t 2
t 1
Ldt;
ð1:75Þ
where H defines the action integral, L is the Lagrangian, t 1 and t 2 are arbitrary
instants of time, and dH is the variation of H.
The principle states that the motion of a mechanical system, from an initial
configuration at time t 1 to a final configuration at time t 2 , occurs in such a manner
that the action integral attains a stationary value with respect to arbitrary admissible
variations of system configurations – provided that the variations of displacements
vanish at times t 1 and t 2 .
At static equilibrium the kinetic energy T vanishes, and the potential energy
V becomes independent of time; Hamilton’s principle then reduces to the principle
of minimum potential energy. Also, one can deduce Lagrange’s equations from
Hamilton’s principle.
Some definitions may now be in order:
Monogenic forces are derivable from a scalar quantity, typically some work or
energy function or potential. For example, the restoring force kx in a linear spring is
monogenic, since one can derive it from the energy function 1/2kx
2 by differentiation.
Forces that cannot be derived from a scalar quantity (dry friction, e.g.) are polygenic.
Holonomic constraints express relations between system coordinates q = q(t)2
R
n in the form f(q, t) = 0, f 2 R
m , i.e. they depend only on position variables and
possibly time, but not on higher time-derivatives such as velocities, and does not
contain inequalities. For example, with transverse beam vibrations u(x,t) the constraints u(0, t) = 0, u″(0, t) = 0, u(0, t) = u(l, t), and u(0, t) = u 0 sin(Xt) are all
holonomic, while u(0, t) ! 0, _
u(0, t) = k _
u(l, t), and _
u(0, t) = Xu 0 cos(Xt) are nonholonomic.
A variation du of a function u is a virtual infinitesimal change of all function
values. This change, by contrast to the infinitesimal d-process of ordinary calculus,
is not caused by an actual change of an independent variable, but is imposed on a
set of dependent variables as a kind of ‘mathematical experiment’. Imagine some
deformation pattern u = u(x). Then change all values of u by slight amounts du(x),
and you have a variation in u(x). An admissible variation make u + du satisfy the
boundary conditions or initial conditions of the problem. Thus, with u specified at
the boundaries, du = 0 at these boundaries for du(x) to be an admissible variation.
Certain rules apply for the calculus of variations of functionals (functions of
functions). For example, the (first) variation dF of a functional F(x, u(x), u′(x),
u″(x), …, u
(n) (x)) is given by a simple chain rule:
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1 Vibration Basics
Hamilton’s principle holds for any mechanical systems subjected to monogenic
forces and holonomic constraints (to be explained). By contrast to Lagrange’s
equations, it applies too for systems characterized by infinitely many degrees of
freedom, that is, for continuous or distributed systems. By Hamilton’s principle,
dH ¼ 0; where H
Z t 2
t 1
Ldt;
ð1:75Þ
where H defines the action integral, L is the Lagrangian, t 1 and t 2 are arbitrary
instants of time, and dH is the variation of H.
The principle states that the motion of a mechanical system, from an initial
configuration at time t 1 to a final configuration at time t 2 , occurs in such a manner
that the action integral attains a stationary value with respect to arbitrary admissible
variations of system configurations – provided that the variations of displacements
vanish at times t 1 and t 2 .
At static equilibrium the kinetic energy T vanishes, and the potential energy
V becomes independent of time; Hamilton’s principle then reduces to the principle
of minimum potential energy. Also, one can deduce Lagrange’s equations from
Hamilton’s principle.
Some definitions may now be in order:
Monogenic forces are derivable from a scalar quantity, typically some work or
energy function or potential. For example, the restoring force kx in a linear spring is
monogenic, since one can derive it from the energy function 1/2kx
2 by differentiation.
Forces that cannot be derived from a scalar quantity (dry friction, e.g.) are polygenic.
Holonomic constraints express relations between system coordinates q = q(t)2
R
n in the form f(q, t) = 0, f 2 R
m , i.e. they depend only on position variables and
possibly time, but not on higher time-derivatives such as velocities, and does not
contain inequalities. For example, with transverse beam vibrations u(x,t) the constraints u(0, t) = 0, u″(0, t) = 0, u(0, t) = u(l, t), and u(0, t) = u 0 sin(Xt) are all
holonomic, while u(0, t) ! 0, _
u(0, t) = k _
u(l, t), and _
u(0, t) = Xu 0 cos(Xt) are nonholonomic.
A variation du of a function u is a virtual infinitesimal change of all function
values. This change, by contrast to the infinitesimal d-process of ordinary calculus,
is not caused by an actual change of an independent variable, but is imposed on a
set of dependent variables as a kind of ‘mathematical experiment’. Imagine some
deformation pattern u = u(x). Then change all values of u by slight amounts du(x),
and you have a variation in u(x). An admissible variation make u + du satisfy the
boundary conditions or initial conditions of the problem. Thus, with u specified at
the boundaries, du = 0 at these boundaries for du(x) to be an admissible variation.
Certain rules apply for the calculus of variations of functionals (functions of
functions). For example, the (first) variation dF of a functional F(x, u(x), u′(x),
u″(x), …, u
(n) (x)) is given by a simple chain rule:
22
1 Vibration Basics