A particle of mass m is attracted to a force center with the force of magnitude k/ r2. Use plane polar coordinates and find Hamilton’s equations of motion.
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A particle of mass m is attracted to a force center with the force of
magnitude k/ r2. Use plane polar coordinates and find Hamilton’s equations
of motion.
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- 6. Consider a mass m which is constrained to move on the frictionless surface of a vertical cone (Fig. 1) (in cylindrical polar coordinates p. p, z with z>0) in a uniform gravitational field g vertically down. Find Hamiltonian function of the cone and deduce Hamilton's equations. z SohCahtoa sing cos r = pcosd + psind Fig. 1Consider a vector field F = 2 pzap+3z sing ap- 4P cosp az. Evaluate the circulation of F over the path (L) defined by 0Express the vector field W=(x^2-y^2)i +xzk in cylindrical coordinates at P(6,60degrees,-4).A vector field is given in spherical coordinates as B = RR² cos(6/2) + Rsin(0) sin(0/2) . Evaluate (V x B) ds over the surface of the lower half of a sphere shown in the figure. Assume the surface normal is  =-R. The parameters are given as: R=7,= 3.14 Note: You may use the Stokes' Theorem. Yanıt: R=a N 40 S y LF(x, y, z) = (x + 1)ỉ – (2y + 1)j+ zk vector field is given. C(1,0,0),(0,1,0),(0,0,1) being the clock wise boundery of the triangle (1,1,1) f Fdr The curvilinear integral. a) By calculating the curved integral b) By stokes theorem calculate.A particle of mass m is forced to move on the inner surface of a smooth cone with side-peak angle α. Particle is under the influence of gravitational force. Find the serialized set of coordinates and obtain the bond equations. Find Lagrange's equations of motion. Note you can select cylindrical coordinatesConsider an ideal gas system composed of Krypton atoms at a temperature of 302.15 K and 116.64 kPa in pressure. The atomic mass of Krypton is 83.8 Da. Given that the total volume of the system is 1 Liter: a. Calculate the total Hamiltonian of the system. b. Calculate the relative fluctuation (coefficient of variation) in the Hamiltonian2. 1D Ising model Consider the Ising model in 1D with zero external field. The Ising Hamiltonian in this case is N-1 #=-JΣ σίσί+1, i=1 where σ = ±1, J > 0. In this problem I am asking to plot some graphs. You can use any graphics software, e.g. gnuplot, excel, etc. (a) (b) (c) (d) Calculate the ground-state energy. Calculate the canonical partition function. Calculate the probability that all spins are aligned. Consider the limit N → ∞. Plot this probability versus kBT/J for few values of N, say N = 103 and N = 108. Argue that your result indicates the absence of ordered state at T > 0. Calculate the Helmholtz free energy A, average energy E, heat capacity C, and the entropy. Plot graphs for A/(JN), E/(JN) and C/N vs kBT/J. In few words discuss your results.Solve the problem of the motion of a point projectile in a vertical plane, using the Hamiltonian-Jacobi method. Find both the equation of the trajectory and the dependence of the coordinates on time, assuming the projectile is fired off at time t = 0 from the origin with the velocity vo, making an angle a with the horizontal.O A projectile is projected at an angle 0 degrees above a horizontal surface with a velocity V, from some origin at timet=0. By using the Hamiltonian method, show that the distance travelled in the y-direction is given by y= v, sin 01-grQ 3: A particle of mass m moves in a plane under the influence of a central force that depends only on its distance from the origin. Write the Hamiltonian and Hamilton’s equations?We can use the calculus of variations to solve a classic problem in the history ofphysics the branchistochrome.Consider a particle moving in a constant force field startingat rest from some point (x1,y1) to some lower point (x2,y2). Find a suitable set of generalized coordinates for point particle moving on the surface of a hemisphere of radius R whose center is at the origin.SEE MORE QUESTIONS