p(x, ¤1, ..., an, B1, . . , Bn),which constitute a general solution of the canonical system (82).94CANONICAL FORM OF THE EULER EQUATIONSCHAP. 4PROBLEMS1. Use the canonical Euler equations to find the extremals of the functionalSVZI + y? VT + ya dx,and verify that they agree with those found in Chap. 1, Prob. 22.Hint. The Hamiltonian isH(x, y, p) = - V + yª – p²,and the corresponding canonical systemdy%3Dhas the first integralp - ya = C?,where C is a constant.2. Consider the action functional= (mx² – xx*) dtcorresponding to a simple harmonic oscillator, i.e., a particle of mass macted upon by a restoring force -xx (cf. Sec. 36.2).system of Euler equations corresponding to J[x], and interpret them. Calcu-late the Poisson brackets [x, p), [x, H] and (p, H). Is p a first integral ofthe canonical Euler equations?Writecanonical3. Use the principle of least action to give a variational formulation of theproblem of the plane motion of a particle of mass m attracted to the originof coordinates by a force inversely proportional to the square of its distancefrom the origin. Write the corresponding equations of motion, the Hamil-tonian and the canonical system of Euler equations. Calculate the Poissonbrackets [r, p,), (0, po), [pr, H] and [Pe, H], whereP, =Po =Is Pe a first integral of the canonical Euler equations?Hint. The action functional isJ[r, 0] =dt,where k is a constant, and r, 0 are the polar coordinates of the particle.4. Verify that the change of variablesY, - Pi,P = yıis a canonical transformation, and find the corresponding generating function.PROBLEMSCANONICAL FORM OF THE EULER EQUATIONS95

Name: p(x, ¤1, ..., an, B1, . . , Bn),which constitute a general solution o
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Description: p(x, ¤1, ..., an, B1, . . , Bn),which constitute a general solution of the canonical system (82).94CANONICAL FORM OF THE EULER EQUATIONSCHAP. 4PROBLEMS1. Use the canonical Euler equations to find the extremals of the functionalSVZI + y? VT + ya dx,a