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(c) The solution to the equations above leads to A(r, y) = 1/cosh°(y) so that ds? (dr²+dy)/ cosh(y). Let (r(t), y(t)) be a path in the plane, parametrised by a variable t (with to

(c) The solution to the equations above leads to A(r, y) = 1/cosh°(y) so that ds? (dr²+dy)/ cosh(y). Let (r(t), y(t)) be a path in the plane, parametrised by a variable t (with to

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3. Cartography involves the projection of the globe onto the two-dimensional plane. Gerardus Mercator in the 16th century created a projection that became standard for maritime maps because it preserves the angles on the sphere, which is crucial for navigation. We shall consider this projection going from the two-dimensional plane to the unit sphere, rather than from the sphere to the plane which would be a bit more complicated. The plane is described by the usual coordinates (x, y) and the unit sphere by the angles (o, X). Here o is the conventional azimuthal angle (providing the longitude) and A is the latitude, related to the standard polar angle 0 through A = 5-0. The figure shows the details. (The shading is used to make the figure visually more accessible and has no further meaning). transformation Angles are preserved if the infinitesimal path length ds on the sphere depends in the same way on dr as on dy, which is true if we are able to write ame ds? = A(x, y)(dx? + dy?). In questions (a) and (b) you will determine A(x, y). In the following questions you must use the properties of conserved Lagrangians to derive the formulae for the shortest path length without solving the Euler-Lagrange equations. (a) Show that ds? on the unit sphere (with radius R = 1, considered for simplicity dimen- sionless) can be written in terms of the spherical angles as ds? = cos (A)do? + dX³. (b) To parametrise ds? in terms of (x, y) we set o = x, and let A = X(y) be a function of y but not of x. Show that then 2 YP A(1r, y) = () dy and that A(y) is the solution of the equation = cos(A(y)). dy %3D
Transcribed Image Text:3. Cartography involves the projection of the globe onto the two-dimensional plane. Gerardus Mercator in the 16th century created a projection that became standard for maritime maps because it preserves the angles on the sphere, which is crucial for navigation. We shall consider this projection going from the two-dimensional plane to the unit sphere, rather than from the sphere to the plane which would be a bit more complicated. The plane is described by the usual coordinates (x, y) and the unit sphere by the angles (o, X). Here o is the conventional azimuthal angle (providing the longitude) and A is the latitude, related to the standard polar angle 0 through A = 5-0. The figure shows the details. (The shading is used to make the figure visually more accessible and has no further meaning). transformation Angles are preserved if the infinitesimal path length ds on the sphere depends in the same way on dr as on dy, which is true if we are able to write ame ds? = A(x, y)(dx? + dy?). In questions (a) and (b) you will determine A(x, y). In the following questions you must use the properties of conserved Lagrangians to derive the formulae for the shortest path length without solving the Euler-Lagrange equations. (a) Show that ds? on the unit sphere (with radius R = 1, considered for simplicity dimen- sionless) can be written in terms of the spherical angles as ds? = cos (A)do? + dX³. (b) To parametrise ds? in terms of (x, y) we set o = x, and let A = X(y) be a function of y but not of x. Show that then 2 YP A(1r, y) = () dy and that A(y) is the solution of the equation = cos(A(y)). dy %3D
(c) The solution to the equations above leads to A(x, y) = 1/cosh°(y) so that ds? (dr? +dy')/ cosh²(y). Let (x(t), y(t)) be a path in the plane, parametrised by a variable t (with to <t < t1) that produces a curve (o, A) on the sphere. With this parametri- sation dr = idt and dy = jdt and the path length e on the sphere is given by the integral - fds = l = dt L(y, i, ý) to with the Lagrangian L(y, r, ý) 1 ² + ÿ². cosh(y) Show that for this Lagrangian one always has aL H = i+ y- - L = 0. (1) (d) To find the equations minimising l it is convenient now to consider the different La- grangian L' = L² instead of L. Generally L' and L would lead to different solutions, but here they are equivalent. As the first step to demonstrate this equivalence, show that L = 0. For this proof you may want to use the conserved first integral for L', i.e. a (i + y - L') = 0, together with H = 0 as given by Eq. (1). Note that you can do this on the formal level without replacing L or L' by their explicit expressions. (e) As the second step show that the Euler-Lagrange equations for L' and L are identical as a consequence of L = 0. Again do this on the formal level without replacing L or L' by their explicit expressions. (f) You have shown above that L and therefore L' is a constant of motion. Show that the canonical momentum p, obtained from L' is another constant of motion. (g) Using the conserved quantities (L', Pz) and the identity = }, show that dy C 1, drV cosh (y) where C is a constant that you should express explicitly in terms of the conserved quantities.
Transcribed Image Text:(c) The solution to the equations above leads to A(x, y) = 1/cosh°(y) so that ds? (dr? +dy')/ cosh²(y). Let (x(t), y(t)) be a path in the plane, parametrised by a variable t (with to <t < t1) that produces a curve (o, A) on the sphere. With this parametri- sation dr = idt and dy = jdt and the path length e on the sphere is given by the integral - fds = l = dt L(y, i, ý) to with the Lagrangian L(y, r, ý) 1 ² + ÿ². cosh(y) Show that for this Lagrangian one always has aL H = i+ y- - L = 0. (1) (d) To find the equations minimising l it is convenient now to consider the different La- grangian L' = L² instead of L. Generally L' and L would lead to different solutions, but here they are equivalent. As the first step to demonstrate this equivalence, show that L = 0. For this proof you may want to use the conserved first integral for L', i.e. a (i + y - L') = 0, together with H = 0 as given by Eq. (1). Note that you can do this on the formal level without replacing L or L' by their explicit expressions. (e) As the second step show that the Euler-Lagrange equations for L' and L are identical as a consequence of L = 0. Again do this on the formal level without replacing L or L' by their explicit expressions. (f) You have shown above that L and therefore L' is a constant of motion. Show that the canonical momentum p, obtained from L' is another constant of motion. (g) Using the conserved quantities (L', Pz) and the identity = }, show that dy C 1, drV cosh (y) where C is a constant that you should express explicitly in terms of the conserved quantities.
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