Classical Solid Mechanics
from the book 'Mechanics' by
Landau and Lifshitz
Principle of Least Action
(Hamiltonian Principle)
Generalized
coordinates completely
define the position of a
system
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Principle of least action
System moves
between t1 and t2
minimizing S.
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General Properties
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Lagrangian equations
→
Additive for noninteracting systems.
Only defined to an
additive total time
derivative →
Assume,
δS = δS', so L and L' give the same result. Time derivative of f
doesn't affect the Lagrangian.
Galilean relativity
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Find frame of
reference such that
space is homogenous
and isotropic and time
is homogenous:
inertial frame.
Lagrangian cannot
explicitly contain r
(not directional) or t.
L must be a function
of v2.
From the Lagrangian equation,
Newton's first law,
Between two inertial frames K and K',
Free particle
Assume inertial frames K and K'.
Taylor expansion,
Second term is either total time derivative or a function of v2.
Define m,
Because the second term is a total time derivative, changing between inertial
reference frames does not have any effect.
System of particles
Since additive, when there is no interaction,
Define kinetic energy,
If interaction between particles,
Newton's second law.
For a single particle moving in an external field,
Conservation of energy
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Comes from the
Lagrangian in a
closed system not
explicitly depending
on time.
Linear function of the
Lagrangian, therefore
additive.
Also valid for systems
with a constant
external field
(conservative field).
Conservation of momentum
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Comes from the
homogeneity of
space.
Assume infinitesimal
change in radius
vector r → r+Є.
Momentum is
additive.
Since Є is arbitrary,
Newton's third law
From Lagrange's equations,
Conservation of angular momentum
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Comes from the
isotropy of space.
Angular momentum is
additive.
Angular momentum difference
between inertial frames
Depends on choice of origin.
Center of mass
Rigid bodies
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Systems of particles
where distances
between particles do
not vary.
Consider center O'
with position a,
velocity v', ang.vel.
Ω'.
Angular velocity the
same for every point.
Inertia tensor
Assume O is center of mass,
Rewrite in tensor form,
Define,
Inertia tensor
Iik is the inertia tensor
Symmetric, so can choose Principle Axes of Inertia by diagonalizing this matrix.
Principle axes of inertia go through CM
Angular momentum of a rigid body
Use CM as the origin O
From definition,
If the axes are the principle axes,
Equations of motion of a rigid body
1. Sum momenta for each particle, For each particle,
For all,
where,
If the system is closed,
Otherwise,
2.
where,
(definition torque, moments)
Rigid bodies in contact
Use constraint equations,
Otherwise use the d'Alembert's principle,
Motion in a non-inertial frame
In an inertial frame K0 with an external field,
Assume a reference frame K' with velocity V(t) relative to K.
Third term can be omitted, also,
One inertial force
Applying the Lagrange's equation,
Take another reference frame K which rotates relative to K'.
Three inertial forces due to rotation (non-uniformity of rotation, Coriolis, centrifugal)
Energies are also different between non-inertial frames