Introduction to Lagrangian & Hamiltonian Mechanics.pdf

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Jacob Linder & Iver H. Brevik
Introduction to Lagrangian &
Hamiltonian Mechanics
2
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Introduction to Lagrangian & Hamiltonian Mechanics
1
st
edition
© 2016 Jacob Linder, Iver H. Brevik &
bookboon.com
ISBN 978-87-403-1249-2
Peer reviewed by Prof. Johan Høye at NTNU, Norway
3
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Introduction to Lagrangian &
Hamiltonian Mechanics
Contents
Contents
Preface
About the authors
I.
A.
B.
C.
D.
E.
F.
G.
II.
A.
B.
C.
Fundamental principles
Notation and brief repetition
Many-particle systems
Constraints and generalized coordinates
D’Alembert’s principle and Lagrange’s equations
Levi-Civita symbol
Friction and other velocity-dependent potentials
Examples
Lagrange’s equations and the variational principle
Hamilton’s principle
Derivation of Lagrange’s equations from Hamilton’s principle
Variational calculus
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Introduction to Lagrangian &
Hamiltonian Mechanics
Contents
D.
E.
III.
A.
B.
IV.
A.
B.
C.
D.
E.
F.
V.
A.
B.
C.
D.
Hamilton’s principle for non-holonomic systems
Conservation laws and symmetries
Hamilton’s equations
Legendre transformations
Going from Lagrangian to Hamiltonian formalism
The two-body problem: central forces
Reduction to equivalent one-body problem
Equations of motion
Equivalent one-dimensional problem
The virial theorem
The Kepler problem
Scattering cross section
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Kinematics and equations of motion for rigid bodies
Orthogonal transformations and independent coordinates
Transformation matrix and its mathematical properties
Formal properties of the transformation matrix
Euler angles
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