Lagrangian Mechanics
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:
A particle is constraint to move in the x-y plane, the equation of constraint is z = 0, the constraint is holonomic. Possible generalized coordinates for the system with two degrees of freedom are x, y; r, φ; … .
A particle is constraint to move on a circle in the x-y plane, the equations of constraints are z = 0, x^{2 }+ y^{2 } - r^{2 }= 0. The constraints are holonomic. Possible generalized coordinates for the system with 1 degree of freedom are φ ; φ^{3}, … .
The Lagrangian of the system is L = T – U. L is a function of the generalized coordinates and velocities and possibly the time, .
If all forces present are conservative or can do no work, them the equations of motion may be obtained from Lagrange’s equations,
The generalized momentum or conjugate momentum or canonical momentum is defined through
Note: If a coordinate q does not explicitly appear in the Lagrangian, then the coordinate is called cyclic and the corresponding conjugate momentum is constant.
The Hamiltonian H of a system is given by Note: H(q, p, t) is a function of the generalized coordinates and momenta and possibly the time.
Hamilton's equations of motion:
Relativity
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})^{-1/2},
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})^{-1/2}.
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})
Example Problems: (Solutions)
Problem 1:
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.
Problem 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.
Problem 3:
Problem 4:
Refer to the previous problem.
Problem 5:
Problem 6:
Problem 7:
Problem 8: Problem 9:Problem 10: