Summary of the concepts you need to be familiar with to solve most GRE problems.
Any set of independent quantities q_{1}, q_{2}, ... , q_{s}, which completely define the position of the system with s degrees of freedom, are called generalized coordinates of the system, and the derivatives are called generalized velocities.
Examples:
The Lagrangian of the system is L = T - U. L is a
function of the generalized coordinates and velocities and possibly the time,
L = L (q, v, t), where q = {q_{i}}, v = {v_{i}}, and v_{i}
= dq_{i}/dt.
If all forces present are conservative or can do no virtual work,
them the equations of motion may be obtained from Lagrange's equations,
d/dt(∂L/∂(dq_{i}/dt)) - ∂L/∂q_{i} = 0.
The generalized momentum or
conjugate momentum or canonical momentum is defined through
p_{i }= ∂L/∂v_{i}.
Note: If a coordinate q_{i} does not explicitly appear in the Lagrangian, then the coordinate is called cyclic and the corresponding conjugate
momentum p_{i} is constant.
The Hamiltonian H of a system is given by H(q, p, t) = ∑_{i}(dq_{i}/dt)p_{i}
- L.
Note: H(q, p, t) is a function of the generalized coordinates and
momenta and possibly the time.
Hamilton's equations of motion: dq_{i}/dt = ∂H/∂p_{i}, dp_{i}/dt = -∂H/∂q_{i}.
Summary of the concepts you need to be familiar with to solve most GRE problems.
I. The laws of nature are the same in all inertial
reference frames.
II. In vacuum, light propagates with respect to any inertial frame and in all
directions with the universal speed c. This speed is a constant of nature.
Proper time τ: The time interval between two events
in a reference frame where the two events have the same space coordinates.
(They "happen" at the same place.)
In a frame moving with speed v with respect to that frame the time interval
between the two events is t = γτ.
γ = (1 - v^{2}/c^{2})^{-½},
t > τ, the proper time interval is the shortest time interval.
Proper length L_{0}: The length (dimension)
of an object in a reference frame in which the object is at rest.
In a frame moving with speed v with respect to that frame the length of the
object in the direction of the relative motion is L = L_{0}/γ. L < L_{0}.
The length of the object in any direction perpendicular to the direction of the
relative motion is the same in both frames.
The space-time interval between two events is ds, ds^{2} = c^{2}dt^{2} - |dr|^{2}. It is a Lorentz invariant quantity, i.e. it is the same in every reference frame.
Relativistic energy and momentum:
E = γmc^{2}, p = γmv, pc/E =
v/c.
E^{2} = m^{2}c^{4} + p^{2}c^{2}.
In every reference frame energy and momentum are conserved.
Lorentz transformation:
Consider two reference frames K and K'. Assume that the coordinate axes in the
two frames are parallel and that the origins of the coordinates coincide at t =
t' = 0. Assume that K' is moving with velocity vi with respect to K. The
Lorentz transformation gives the coordinates of a
space-time point (x_{0},x_{1},x_{2},x_{3})
= (ct,x,y,z) in K in terms of its coordinates (x'_{0},x'_{1},x'_{2},x'_{3})
= (ct',x',y', z') in K' and vice versa.
.
β = v/c, β = v/c, γ = (1 - β^{2})^{-½}.
Velocity addition:
A particle moves in K with
velocity u = dr/dt.
K' moves with respect to K with velocity v.
The particle's velocity in K', u' = dr'/dt', is given by
u'_{||} = (u_{||} - v)/(1 - v∙u/c^{2}),
u'_{⊥} = u_{⊥}/(γ(1 - v∙u/c^{2})).
A non-relativistic particle of mass m moves in a plane. Its position is described by the polar coordinates r and θ. There exists a potential energy U = kr^{2}, where k is a constant. Which of the following is the Lagrangian of the particle?
(A) L = ½m[(dr/dt)^{2} + r^{2}(dθ/dt)^{2}] - kr^{2}
(B) L = ½m[(dr/dt)^{2} + θ^{2}] + kr^{2}.
(C) L = ½m[θ^{2}(dr/dt)^{2} + r^{2}(dθ/dt)^{2}] - kr^{2}.
(D) L = ½m[(dr/dt)^{2} + r(dr/dt)(dθ/dt) + r^{2}(dθ/dt)^{2}] - kr^{2}.
(E) L = ½m[(dr/dt)^{2} + 2r(dr/dt)(dθ/dt) + r^{2}(dθ/dt)^{2}] - kr^{2}.
A non-relativistic particle of mass m moves in a plane. Its position is described by the polar coordinates r and θ. There exists a potential energy U = kr^{2}, where k is a constant. Which of the following quantities remains constant?
(A) m[(dr/dt)^{2} + r^{2}(dθ/dt)^{2}]
(B) mr^{2}(dθ/dt)^{2}
(C) kr^{2
}(D) mr^{2}dθ/dt)
(E) mr^{2}(dθ/dt)
The Lagrangian for a system with generalized coordinate q is L = m(dq/dt)^{4} -g(q), where g(q) is an arbitrary function of the coordinate. The canonical momentum conjugate to q is
(A) m(dq/dt) (B) m(dq/dt)^{3} (C) g(q)(dq/dt) (D) m(dq/dt)^{2}/3 (E) 4m(dq/dt)^{3}
Refer to the previous problem.
Which of the following is a constant of motion for this system?
(A) ½ m(dq/dt)^{2} + g(q) (B)
m(dq/dt)^{2} + g(q) (C) m(dq/dt)^{4}
+ g(q)
(D) 3 m(dq/dt)^{4} + g(q)
(E) 8 m^{2}(dq/dt)^{6} + g(q)
In an inertial frame S a particle has momentum (p_{x}, p_{y}, p^{z}) = (5, 3, √2) MeV/c and a total energy E = 10 MeV. The speed of the particle as measured in frame S is most nearly
(A) (3/8) c (B) (2/5) c (C) ½ c (D) (3/5) c (E) (4/5) c
Which of the following combinations of momentum p' and energy E' could represent the motion of the particle described in the previous problem as observed in another inertial frame S' moving with an unspecified velocity v relative to S.
(A) p' = (0, 0, 8) MeV/c, E' = √(128) MeV
(B) p' = (8, 0, √2) MeV/c, E' = 10 MeV
(C) p' = (31, 4, 6) MeV/c, E' = √(949) MeV
(D) p' = (50, -30, √(200)) MeV/c, E' = 100 MeV
(E) p' = (100, 100, 0) MeV/c, E' = 10,000 MeV
The percentage increase in the energy of a particle whose speed changes from rest to 0.8 c is closest to
(A) 50% (B) 67% (C) 75% (D) 80% (E) 88%
Two spaceships, each measuring 100 m in length in its own rest frame, pass by each other traveling in opposite directions. Instruments on spaceship 1 determine that the front end of spaceship 1 requires (5/3)*10^{-7 }s to traverse the full length of spaceship 2. What is the relative speed of the two ships?
(A) (1/√6)c (B) (½)c (C) (1/√2)c (D) (2/√5)c (E) (2/√3)c
A proton has kinetic energy 500 GeV. The momentum of the proton is most nearly
(A) 22 GeV/c (B) 30 GeV/c (C) 250 GeV/c (D) 500 GeV/c (E) 707 GeV/c
A ∑^{0} particle (ass M_{1}) decays at rest in the laboratory into a Λ^{0} particle (mass M_{2}) and a massless photon. The energy of the Λ^{0} particle is
(A) M_{1}c^{2}/2 (B) (M_{1}^{2} + M_{2}^{2})c^{2}/(2M_{1}) (C) (M_{1} + M_{2})^{2}c^{2}/(2M_{1})
(D) (M_{1}^{2} - M_{2}^{2})c^{2}/(2M_{1}) (E) (M_{1} + M_{2})c^{2}/2