Lagrangian and Hamiltonian mechanics
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Lagrangian and Hamiltonian mechanics
The Lagrangian for a particle with velocity i = y and electric...
Conservation of energy: Using Hamiltonian or Lagrangian Mechanics 2) A particle P, of mass m, is...
Conservation of energy: Using Hamiltonian or Lagrangian Mechanics 2) A particle P, of mass m, is attached by means of two light ideal springs (no damping) to fixed points A and B such that APB is a vertical straight line of length 5a. Spring AP is of stiffness k, spring PB is of stiffness 4k, and both springs are of natural length a. Point A is directly above B. i) Show that when the particle is in equilibrium AP =...
For a charged particle (with charge e) in an electromagnetic field the Hamiltonian can be written...
For a charged particle (with charge e) in an electromagnetic field the Hamiltonian can be written as: 1 e H (inő A) +eº (2) 2m where A is the vector potential and o is the scalar potential of the field. a) Find the form of the operator for the velocity, v, of a charged particle in an electromagnetic field. Hint: try working this out for a single component (say the x-component) and then generalize. b) Is the velocity a simultaneous...
19. Consider a charged particle (charge q) at the origin surrounded by an infinitely long cylindrical...
19. Consider a charged particle (charge q) at the origin surrounded by an infinitely long cylindrical shell (radius a) of a magnetic field in the φ direction, i.e. B = Ao δ[s- (a) Determine the electromagnetic momentum density g-60 EB. (b) Determine the total momentum in the field by integrating over all space. (c) Determine the vector potential corresponding to the magnetic field and show that the electromagnetic momentum is just the vector potential at the position of the charge...
3. The Lagrangian for a relativistic particle of (rest) mass m is L=-me²/1- (A² - Elmo...
3. The Lagrangian for a relativistic particle of (rest) mass m is L=-me²/1- (A² - Elmo (The corresponding action S = ( L dt is simply the length of the particle's path through space-time.) (a) Show that in the nonrelativistic limit (v << c) the result is the correct nonrelativistic kinetic energy, plus a constant corresponding to the particle's rest energy. (Hint. Use the binomial expansion: for small 2, (1 + 2) = 1 +a +a(-1) + a(a-1)(-2) 13 +...
PROBLEM 2 1. The Lagrangian L for a single particle in a potential V(2) in one...
PROBLEM 2
1. The Lagrangian L for a single particle in a potential V(2) in one dimension is [(4,) = 5 mi? – V(x) Find the Hamiltonian H defined H2,p) = max{sip - L(r,.ic)} How does this compare to the total energy? 2. Repeat the previous problem for a single particle in a potential V (2, y, z) in three dimensions by doing a Legendre transform of the i, followed by one for y, and lastly one for 2.
wCT S the particle's 4-velocity 12.13 Specalize the Darwin Lagrangian (12.82) to the interaction of two...
wCT S the particle's 4-velocity 12.13 Specalize the Darwin Lagrangian (12.82) to the interaction of two charged particles (m, q) and (m, qa). Introduce reduced particle coordinates X1X2, V -2 and also center of mass coordinates. Write out the Lagrangian in the reference frame in which the velocity of the center of mass vanishes and evaluate the canonical momentum components, p, aLlau (a) r = etc. action is known as the Breit interaction (1930). For system of interacting charged particles...
# Problem 1 # Suppose a point-mass particle with mass, 'm', moving in a gravitational potential,...
# Problem 1 # Suppose a point-mass particle with mass, 'm', moving in a gravitational potential, 'U(r)', where 'r' is the distance from the center of the potential. A positional vector and momentum vector of a particle are vec r' and "vec p', respectively. (\vec means vector symbol.) Q1) An angular momentum vector vec J' is defined as vec J = \vec r x \vec p. Show that \vec J is conserved in such a gravitational potential U(r) which depends...
Please provide clear steps, I'm a little confused about how to solve this question. Thanks. Problem...
Please provide clear steps, I'm a little confused about how to
solve this question. Thanks.
Problem 2. (20 points) Particle in a magnetic field. The Hamiltonian for a particle of mass m and charge e in arn electronnagnetic field with scalar potential φ(r,t) and vector potential . (r,t) is given by where φ is the scalar potential and A is the vector potential. 2.1. (10 points) Show that the following transformation is canonical for any choice of the parameter o...
1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at ti...
Mechanics. Need help with c) and d)
1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at time t. The particle has potential energy V(x, y, 2) so that its Lagrangian is given by where i d/dt, dy/dt, dz/dt (a) Writing q(q2.93)-(r, y, z) and denoting by p (p,P2, ps) their associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy) H(q,p)H(g1, 92,9q3,...
9. An electron moving with non-relativistic velocity v in an electric field E experiences a magnetic...
9. An electron moving with non-relativistic velocity v in an electric field E experiences a magnetic fieldB given by: v x (-V(r)) v x E B=- where (r) is the electric potential. This magnetic field interacts with the magnetic moment u of the electron given by -S, =n me where S is the electron spin. Assuming non-relativistic mechanics, show that the Hamiltonian representing this effect (spin-orbit coupling) for a spherically-symmetric electric potential is: 1 dφ(r) S.L ΔΗ [6] r dr...
Conservation of energy: Using Hamiltonian or Lagrangian Mechanics 2) A particle P, of mass m, is attached by means of two light ideal springs (no damping) to fixed points A and B such that APB is a vertical straight line of length 5a. Spring AP is of stiffness k, spring PB is of stiffness 4k, and both springs are of natural length a. Point A is directly above B. i) Show that when the particle is in equilibrium AP =...
For a charged particle (with charge e) in an electromagnetic field the Hamiltonian can be written as: 1 e H (inő A) +eº (2) 2m where A is the vector potential and o is the scalar potential of the field. a) Find the form of the operator for the velocity, v, of a charged particle in an electromagnetic field. Hint: try working this out for a single component (say the x-component) and then generalize. b) Is the velocity a simultaneous...
19. Consider a charged particle (charge q) at the origin surrounded by an infinitely long cylindrical shell (radius a) of a magnetic field in the φ direction, i.e. B = Ao δ[s- (a) Determine the electromagnetic momentum density g-60 EB. (b) Determine the total momentum in the field by integrating over all space. (c) Determine the vector potential corresponding to the magnetic field and show that the electromagnetic momentum is just the vector potential at the position of the charge...
3. The Lagrangian for a relativistic particle of (rest) mass m is L=-me²/1- (A² - Elmo (The corresponding action S = ( L dt is simply the length of the particle's path through space-time.) (a) Show that in the nonrelativistic limit (v << c) the result is the correct nonrelativistic kinetic energy, plus a constant corresponding to the particle's rest energy. (Hint. Use the binomial expansion: for small 2, (1 + 2) = 1 +a +a(-1) + a(a-1)(-2) 13 +...
PROBLEM 2
1. The Lagrangian L for a single particle in a potential V(2) in one dimension is [(4,) = 5 mi? – V(x) Find the Hamiltonian H defined H2,p) = max{sip - L(r,.ic)} How does this compare to the total energy? 2. Repeat the previous problem for a single particle in a potential V (2, y, z) in three dimensions by doing a Legendre transform of the i, followed by one for y, and lastly one for 2.
wCT S the particle's 4-velocity 12.13 Specalize the Darwin Lagrangian (12.82) to the interaction of two charged particles (m, q) and (m, qa). Introduce reduced particle coordinates X1X2, V -2 and also center of mass coordinates. Write out the Lagrangian in the reference frame in which the velocity of the center of mass vanishes and evaluate the canonical momentum components, p, aLlau (a) r = etc. action is known as the Breit interaction (1930). For system of interacting charged particles...
# Problem 1 # Suppose a point-mass particle with mass, 'm', moving in a gravitational potential, 'U(r)', where 'r' is the distance from the center of the potential. A positional vector and momentum vector of a particle are vec r' and "vec p', respectively. (\vec means vector symbol.) Q1) An angular momentum vector vec J' is defined as vec J = \vec r x \vec p. Show that \vec J is conserved in such a gravitational potential U(r) which depends...
Please provide clear steps, I'm a little confused about how to
solve this question. Thanks.
Problem 2. (20 points) Particle in a magnetic field. The Hamiltonian for a particle of mass m and charge e in arn electronnagnetic field with scalar potential φ(r,t) and vector potential . (r,t) is given by where φ is the scalar potential and A is the vector potential. 2.1. (10 points) Show that the following transformation is canonical for any choice of the parameter o...
Mechanics. Need help with c) and d)
1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at time t. The particle has potential energy V(x, y, 2) so that its Lagrangian is given by where i d/dt, dy/dt, dz/dt (a) Writing q(q2.93)-(r, y, z) and denoting by p (p,P2, ps) their associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy) H(q,p)H(g1, 92,9q3,...
9. An electron moving with non-relativistic velocity v in an electric field E experiences a magnetic fieldB given by: v x (-V(r)) v x E B=- where (r) is the electric potential. This magnetic field interacts with the magnetic moment u of the electron given by -S, =n me where S is the electron spin. Assuming non-relativistic mechanics, show that the Hamiltonian representing this effect (spin-orbit coupling) for a spherically-symmetric electric potential is: 1 dφ(r) S.L ΔΗ [6] r dr...
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