Physics 200A Theoretical Mechanics Fall 2013 Topics

... Energy theorem, wave energy and momentum, wave momentum evolution d.) Symmetry in continuum dynamics e.) D’Alembert’s solution to wave equation, separation f.) Basic Ideas of Fluids g.) Simple Ideas in Potential and Viscous Flow ...

... Energy theorem, wave energy and momentum, wave momentum evolution d.) Symmetry in continuum dynamics e.) D’Alembert’s solution to wave equation, separation f.) Basic Ideas of Fluids g.) Simple Ideas in Potential and Viscous Flow ...

Classical Mechanics and Minimal Action

... is minimal. The quantity S is referred to as the Action and the integrand L(q, q̇, t) the Lagrangian. The Lagrangian of a physical system is defined to be the difference between kinetic- and potential energy. That is, if T is the kinetic energy and V the potential energy, then L = T − V. The princip ...

... is minimal. The quantity S is referred to as the Action and the integrand L(q, q̇, t) the Lagrangian. The Lagrangian of a physical system is defined to be the difference between kinetic- and potential energy. That is, if T is the kinetic energy and V the potential energy, then L = T − V. The princip ...

PHYS4330 Theoretical Mechanics HW #8 Due 25 Oct 2011

... (3) (See Taylor 7.49.) A mass m with charge q moves in a uniform constant magnetic field B = Bẑ. Prove that B = ∇ × A where A = 12 B × r. (You can do this in a coordinateindependent way, only assuming that B is a constant field, and using some vector identities.) Show that A = 12 Bρφ̂ in cylindrica ...

... (3) (See Taylor 7.49.) A mass m with charge q moves in a uniform constant magnetic field B = Bẑ. Prove that B = ∇ × A where A = 12 B × r. (You can do this in a coordinateindependent way, only assuming that B is a constant field, and using some vector identities.) Show that A = 12 Bρφ̂ in cylindrica ...

x,

... Using the method of reduction of order, find another linear independent solution. 2. Using Laplace transform solve the boundary value problem y"- 2yf+ y(x) = x, ...

... Using the method of reduction of order, find another linear independent solution. 2. Using Laplace transform solve the boundary value problem y"- 2yf+ y(x) = x, ...

The Two-Body problem

... Now consider symmetries. First, homogeneity of space implies conservation of the total linear momentum, that is Ṙ is constant - the CoM moves with constant velocity. Of course, we can deduce this by observing that R is ignorable/cyclic coordinate in the Lagrangian. (In one of the example questions ...

... Now consider symmetries. First, homogeneity of space implies conservation of the total linear momentum, that is Ṙ is constant - the CoM moves with constant velocity. Of course, we can deduce this by observing that R is ignorable/cyclic coordinate in the Lagrangian. (In one of the example questions ...

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

... 7. What is the nature of the path traced by a representative point in a two dimensional phase space for a one dimensional harmonic oscillator? 8. What is the nature of the new set of variables ( transformation from the set of variables ( , ) to ( , is zero? 9. What are coupled oscillators? ...

... 7. What is the nature of the path traced by a representative point in a two dimensional phase space for a one dimensional harmonic oscillator? 8. What is the nature of the new set of variables ( transformation from the set of variables ( , ) to ( , is zero? 9. What are coupled oscillators? ...

Solution to problem 2

... as Σ B · dn = 0 (thanks to the Stokes’ theorem), which means that the net flux of the magnetic field through any closed surface Σ is always zero; in other words, H there are no!magnetic monopoles. The second one, ∇ × E + ∂t B = 0, is the Faraday’s law of induction, in the integral form, ∂Σ E · dl = ...

... as Σ B · dn = 0 (thanks to the Stokes’ theorem), which means that the net flux of the magnetic field through any closed surface Σ is always zero; in other words, H there are no!magnetic monopoles. The second one, ∇ × E + ∂t B = 0, is the Faraday’s law of induction, in the integral form, ∂Σ E · dl = ...

Noether's (first) theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proven by German mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action.Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. A generalization of the seminal formulations on constants of motion in Lagrangian and Hamiltonian mechanics (developed in 1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian alone (e.g. systems with a Rayleigh dissipation function). In particular, dissipative systems with continuous symmetries need not have a corresponding conservation law.