Explanation Calculation: Consider the force acted on the motion of the particle: F = − ∇ V , Here V is the potential energy. The objective is to write the Lagrange’s equation for the motion of a particle using spherical coordinates. The element of arc length in spherical coordinates is d s where d s 2 = d r 2 + r 2 d θ + r 2 sin 2 θ d ϕ 2 If s represents the distance the particle has moved along some path, then d s d t is the velocity of the particle. Divide d s 2 = d r 2 + r 2 d θ + r 2 sin 2 θ d ϕ 2 by d t 2 , then the square of the velocity is as follows: V 2 = ( d s d t ) 2 = ( d r d t ) 2 + r 2 ( d θ d t ) 2 + r 2 sin 2 θ ( d ϕ d t ) 2 = r ˙ 2 + r 2 θ ˙ 2 + r 2 sin 2 θ ϕ ˙ 2 The kinetic energy is obtained using the formula T = 1 2 m v 2 . Substitute V 2 in T = 1 2 m v 2 . T = 1 2 m [ r ˙ 2 + r 2 θ ˙ 2 + r 2 sin 2 θ ϕ ˙ 2 ] The motion of the particle is obtained by subtracting potential energy from kinetic energy. L = T − V = 1 2 m [ r ˙ 2 + r 2 θ ˙ 2 + r 2 sin 2 θ ϕ ˙ 2 ] − V ( r , θ , ϕ ) The lagranges equation in the variables r , θ and ϕ are: d d t ∂ L ∂ r ˙ − ∂ L ∂ r = 0 d d t ∂ L ∂ θ ˙ − ∂ L ∂ θ = 0 d d t ∂ L ∂ ϕ − ∂ L ∂ ϕ = 0 Substitute L into the Lagranges equations. d d t ( m r ˙ ) − m θ ˙ 2 r + m r sin 2 θ ϕ 2 ∂ V ∂ r = 0 d d t ( m r 2 θ ˙ ) − m r 2 sin θ cos θ ϕ 2 + ∂ V ∂ θ = 0 d d t ( m r sin 2 θ ϕ ˙ 2 ) + ∂ V ∂ θ = 0 The r equation of motion is as follows: m ( r ¨ − r θ ˙ 2 − r ϕ ˙ 2 sin 2 θ ) = − ∂ V ∂ t ...... ( 1 ) The θ equation of motion is as follows: m ( 2 r ˙ θ ˙ + r θ ¨ − r ϕ ˙ 2 sin θ cos θ ) = − 1 r ∂ V ∂ θ

Mathematical Methods in the Physical Sciences

3rd Edition

ISBN: 9780471198260

Author: Mary L. Boas

Publisher: Wiley, John & Sons, Incorporated

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Cylindrical Coordinate System

The distance from a selected reference axis, the direction from the axis relative to a selected reference direction, and also the distance from a selected reference plane perpendicular to the axis area unit, all accustom outline purpose positions in a cy…

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Chapter 10.9, Problem 4P

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To Write: the LaGrange equations for the motion of a particle using spherical coordinates and find the components of the acceleration and compare the results of problem 3

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Chapter 10.9, Problem 3P

Chapter 10.9, Problem 5P

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Mathematical Methods in the Physical Sciences

Ch. 10.2 - Verify equations (2.6).Ch. 10.2 - Prob. 2P Ch. 10.2 - Consider the matrix A in (2.7) or (2.10). Think of...Ch. 10.2 - Any rotation of axes in three dimensions can be...Ch. 10.2 - Write equations (2.12) out in detail and solve the...Ch. 10.2 - Write the transformation equation for a 3rd-rank...Ch. 10.2 - Following what we did in equations (2.14) to...Ch. 10.2 - Write the equations in (2.16) and so in (2.17)...Ch. 10.3 - Write equations (2.11,), (2.12), (2.13), (2.14),...Ch. 10.3 - Show that the fourth expression in (3.1) is equal...

Ch. 10.3 - As we did in (3.3), show that the contracted...Ch. 10.3 - Show that the contracted tensor TijkVk is a 2nd...Ch. 10.3 - Show that TijklmSlm is a tensor and find its rank...Ch. 10.3 - Show that the sum of two 3rd -rank tensors is a...Ch. 10.3 - As in problem 6, show that the sum of two 2nd...Ch. 10.3 - Show that (3.9) follows from (3.8). Hint: Give a...Ch. 10.3 - Prove the quotient rule in each of the following...Ch. 10.3 - Prove the quotient rule in each of the following...Ch. 10.3 - Prove the quotient rule in each of the following...Ch. 10.3 - Prove the quotient rule in each of the following...Ch. 10.3 - Show that the first parenthesis in (3.5) is a...Ch. 10.4 - As in (4.3) and (4.4), find the y and z components...Ch. 10.4 - Complete Example 4 to verify the rest of the...Ch. 10.4 - As in Problem 2, complete Example 5.Ch. 10.4 - Find the inertia tensor about the origin for a...Ch. 10.4 - For the mass distributions in Problems 5 to 7,...Ch. 10.4 - For the mass distributions in Problems 5 to 7,...Ch. 10.4 - For the mass distributions in Problems 5 to 7,...Ch. 10.4 - For the mass distributions in Problems 5 to 7,...Ch. 10.5 - Verify that (5.5) agrees with a Laplace...Ch. 10.5 - Verify for a few representative cases that (5.6)...Ch. 10.5 - Show that ijklm is an isotropic tensor of rank...Ch. 10.5 - Generalize Problem 3 to see that the direct...Ch. 10.5 - Let Tjkmn be the tensor in (5.8). This is a...Ch. 10.5 - Evaluate: (a) ijjkkmim (b) ijkjk (c) jk2k2j (d)...Ch. 10.5 - Write in terms of s as in (5.8) and (5.9): (a)...Ch. 10.5 - Show that the equations (5.10) are correct. Hints:...Ch. 10.5 - (a) Finish the work of showing that the cross...Ch. 10.5 - (a) Write the triple scalar product A(BC) in...Ch. 10.5 - Using problem 10, write A(BA) in tensor notation...Ch. 10.5 - Write and prove in tensor notation: (a) Chapter 6,...Ch. 10.5 - Write in tensor notation and prove the following...Ch. 10.5 - Show that the diagonal elements of an...Ch. 10.5 - Write a 4-by-4 antisymmetric matrix to show that...Ch. 10.5 - Verify that (5.16) gives (5.17). Also verify that...Ch. 10.5 - Write out the components of Tjk=AjBkAkBj to show...Ch. 10.6 - Show that in 2 dimension (say the x, y plane), an...Ch. 10.6 - In Chapter 3, we said that any 3-by-3 orthogonal...Ch. 10.6 - For Example 1, write out the components of U,V,...Ch. 10.6 - Do Example 1 and Problem 3 if the transformation...Ch. 10.6 - Write the tensor transformation equations for...Ch. 10.6 - Prob. 6P Ch. 10.6 - Write the transformation equations for the triple...Ch. 10.6 - Write the transformation equations for WS to...Ch. 10.6 - Prob. 9P Ch. 10.6 - Prob. 10P Ch. 10.6 - Prob. 11P Ch. 10.6 - Prob. 12P Ch. 10.6 - Prob. 13P Ch. 10.6 - Prob. 14P Ch. 10.6 - In equation (5.12), find whether A(BC) is a vector...Ch. 10.6 - In equation (5.14), is (V) a vector or a...Ch. 10.6 - In equation (5.16), show that if Tjk is a tensor...Ch. 10.7 - Verify (7.1).Hints: In Figure 7.1, consider the...Ch. 10.7 - Write out the sums Pijej for each value of i and...Ch. 10.7 - Carry through the details of getting (7.4) from...Ch. 10.7 - Interpret the elements of the matrices in Chapter...Ch. 10.7 - Show by the quotient rule (Section 3) that Cijkm...Ch. 10.7 - If P and S are 2nd-rank tensors, show that 92=81...Ch. 10.7 - In (7.9) we have written the first row of elements...Ch. 10.7 - Do Problem 4.8 in tensor notation and compare the...Ch. 10.8 - Find ds2 in spherical coordinates by the method...Ch. 10.8 - Observe that a simpler way to find the velocity...Ch. 10.8 - Prob. 3P Ch. 10.8 - In the text and problems so far, we have found the...Ch. 10.8 - Prob. 5P Ch. 10.8 - As in Problem 1, find ds2, the scale factors, the...Ch. 10.8 - As in Problem 1, find ds2, the scale factors, the...Ch. 10.8 - As in Problem 1, find ds2, the scale factors, the...Ch. 10.8 - As in Problem 1, find ds2, the scale factors, the...Ch. 10.8 - Sketch or computer plot the coordinate surfaces in...Ch. 10.8 - Prob. 11P Ch. 10.8 - Using the expression you have found for ds, and...Ch. 10.8 - Prob. 13P Ch. 10.8 - Using the expression you have found for ds, and...Ch. 10.8 - Let x=u+v,y=v. Find ds, thea vectors, and ds2 for...Ch. 10.9 - Prove (9.4) in the following way. Using (9.2) with...Ch. 10.9 - Prob. 2P Ch. 10.9 - Using cylindrical coordinates write the Lagrange...Ch. 10.9 - Prob. 4P Ch. 10.9 - Write out U,V,2U, and V in spherical coordinates.Ch. 10.9 - Do Problem 3 for the coordinate systems indicated...Ch. 10.9 - Do Problem 3 for the coordinate systems indicated...Ch. 10.9 - Do Problem 3 for the coordinate systems indicated...Ch. 10.9 - Do Problem 3 for the coordinate systems indicated...Ch. 10.9 - Do Problem 5 for the coordinate systems indicated...Ch. 10.9 - Do Problem 5 for the coordinate systems indicated...Ch. 10.9 - Do Problem 5 for the coordinate systems indicated...Ch. 10.9 - Do Problem 5 for the coordinate systems indicated...Ch. 10.9 - Prob. 14P Ch. 10.9 - Prob. 15P Ch. 10.9 - Use equations (9.2), (9.8), and (9.11) to evaluate...Ch. 10.9 - Use equations (9.2), (9.8), and (9.11) to evaluate...Ch. 10.9 - Use equations (9.2), (9.8), and (9.18) to evaluate...Ch. 10.9 - Use equations (9.2), (9.8) and (9.11) to evaluate...Ch. 10.9 - Use equations (9.2), (9.8), and (9.11) to evaluate...Ch. 10.9 - Use equations (9.2), (9.8), and (9.11) to evaluate...Ch. 10.10 - Verify equation (10.7). Hint: Use equations (2.4)...Ch. 10.10 - From (10.1) find /x=(1/r)coscos and show that...Ch. 10.10 - Divide equation (10.4) by dt to show that the...Ch. 10.10 - Prob. 4P Ch. 10.10 - Write u in polar coordinates in terms of its...Ch. 10.10 - Prob. 6P Ch. 10.10 - As in (10.12), write the transformation equations...Ch. 10.10 - Using (10.15) show that gij is a 2nd-rank...Ch. 10.10 - If Ui is a contravariant vector and Vj is a...Ch. 10.10 - Show that if Vi is a contravariant vector then...Ch. 10.10 - In (10.18), show by raising and lowering indices...Ch. 10.10 - Show that in a general coordinate system with...Ch. 10.10 - Verify (10.20).Ch. 10.10 - Using equations (10.20) to (10.23), write the...Ch. 10.10 - Do Problem 14 for an orthogonal coordinate system...Ch. 10.10 - Continue Problem 8.15 to find the gij matrix and...Ch. 10.10 - Repeat Problems 8.15 and 10.16 above for the (u,v)...Ch. 10.10 - Using (10.19), show that aiai=ji.Ch. 10.11 - Show that the transformation equation for a...Ch. 10.11 - Let e1,e2,e3 be a set of orthogonal unit vectors...Ch. 10.11 - In Chapter 3, Problem 6.6, you are asked to prove...Ch. 10.11 - If E= electric field and B= magnetic field, is EB...Ch. 10.11 - Do Problems 5 to 8 for the (u,v) coordinate system...Ch. 10.11 - Do Problems 5 to 8 for the (u,v) coordinate system...Ch. 10.11 - Do Problems 5 to 8 for the (u,v) coordinate system...Ch. 10.11 - Do Problems 5 to 8 for the (u,v) coordinate system...Ch. 10.11 - If u is a vector specifying the displacement under...Ch. 10.11 - Show that elements Rij of a rotation matrix are...Ch. 10.11 - Show that the nine quantities Tij=Vi/xj (which are...Ch. 10.11 - The square matrix in equation (10.3) is called the...Ch. 10.11 - In equation (10.13) let the x variables be...

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