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- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/04%3A_Nonlinear_Systems_and_Chaos/4.02%3A_Weak_NonlinearityMost physical oscillators become non-linear with increase in amplitude of the oscillations. Consequences of non-linearity include breakdown of superposition, introduction of additional harmonics, and ...Most physical oscillators become non-linear with increase in amplitude of the oscillations. Consequences of non-linearity include breakdown of superposition, introduction of additional harmonics, and complicated chaotic motion that has great sensitivity to the initial conditions as illustrated in this chapter. Weak non-linearity is interesting since perturbation theory can be used to solve the non-linear equations of motion.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/02%3A_Review_of_Newtonian_Mechanics/2.08%3A_Total_Linear_Momentum_of_a_Many-body_SystemCenter of mass.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/18%3A_The_Transition_to_Quantum_Physics/18.01%3A_Introduction_to_Quantum_PhysicsQuantum mechanics supersedes classical mechanics as the fundamental theory of mechanics. Hamiltonian mechanics provided the foundation upon which quantum mechanics was built.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/03%3A_Linear_Oscillators/3.E%3A_Linear_Oscillators_(Exercises)An unusual pendulum is made by fixing a string to a horizontal cylinder of radius R, wrapping the string several times around the cylinder, and then tying a mass m to the loose end. Obtain the...An unusual pendulum is made by fixing a string to a horizontal cylinder of radius R, wrapping the string several times around the cylinder, and then tying a mass m to the loose end. Obtain the equation of motion, and in the approximation sinθ≈θ, show that the natural frequency is ω0=√ql, where g is the gravitational field strength.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/05%3A_Calculus_of_Variations/5.04%3A_Selection_of_the_Independent_VariableA wide selection of variables can be chosen as the independent variable for variational calculus. Selecting which variable to use as the independent variable does not change the physics of a problem, ...A wide selection of variables can be chosen as the independent variable for variational calculus. Selecting which variable to use as the independent variable does not change the physics of a problem, but some selections can simplify the mathematics for obtaining an analytic solution. The following example of a cylindrically-symmetric soap-bubble surface formed by blowing a soap bubble that stretches between two circular hoops, illustrates the importance of the independent variable.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/15%3A_Advanced_Hamiltonian_Mechanics/15.S%3A_Advanced_Hamiltonian_mechanics_(Summary)It has been shown that the Lagrangian and Hamiltonian formulations represent the vector force fields, and the corresponding equations of motion, in terms of the Lagrangian function \(L(\mathbf{q}, \ma...It has been shown that the Lagrangian and Hamiltonian formulations represent the vector force fields, and the corresponding equations of motion, in terms of the Lagrangian function L(q,˙q,t), or the action functional S(q,p,t), which are scalars under rotation.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/02%3A_Review_of_Newtonian_Mechanics/2.09%3A_Angular_Momentum_of_a_Many-Body_SystemFor a many-body system it is possible to separate the angular momentum into two components. One component is the angular momentum about the center of mass and the other component is the angular motion...For a many-body system it is possible to separate the angular momentum into two components. One component is the angular momentum about the center of mass and the other component is the angular motion of the center of mass about the origin of the coordinate system. This separation is done by describing the angular momentum of a many-body system using a position vector with respect to the center of mass plus the vector location of the center of mass.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/03%3A_Linear_Oscillators/3.07%3A_Wave_equationWave motion is a ubiquitous feature in nature. Mechanical wave motion is manifest by transverse waves on fluid surfaces, longitudinal and transverse seismic waves travelling through the Earth, and vib...Wave motion is a ubiquitous feature in nature. Mechanical wave motion is manifest by transverse waves on fluid surfaces, longitudinal and transverse seismic waves travelling through the Earth, and vibrations of mechanical structures such as suspended cables.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/03%3A_Linear_Oscillators/3.06%3A_Sinusoidally-driven_linearly-damped_linear_oscillatorThis occurs frequently in nature.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/17%3A_Relativistic_Mechanics/17.04%3A_Relativistic_KinematicsRelative differences between Newtonian and relativistic kinematics.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/zz%3A_Back_Matter/20%3A_GlossaryThe moment of inertia of a plane lamina body about, an axis perpendicular to the plane of the lamina, is equal to the sum of the moments of inertia of the lamina about two axes at right angles to each...The moment of inertia of a plane lamina body about, an axis perpendicular to the plane of the lamina, is equal to the sum of the moments of inertia of the lamina about two axes at right angles to each other, in its own plane intersecting each other at a point where the perpendicular axis passes through it.
