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We consider an object of mass m moving in one spatial dimension. We denote its position by r(t) in a given system of coordinates where t represents the time. With the notations used in the course, we write the Lagrangian of the system as 1 L(x,x) = -mx² – V(x), where the potential energy V(r) does not depend explicitly on any variables other than x. In the lectures we have seen that V(x) is defined through F(x) = == ƏV (x) əx 2

We consider an object of mass m moving in one spatial dimension. We denote its position by r(t) in a given system of coordinates where t represents the time. With the notations used in the course, we write the Lagrangian of the system as 1 L(x,x) = -mx² – V(x), where the potential energy V(r) does not depend explicitly on any variables other than x. In the lectures we have seen that V(x) is defined through F(x) = == ƏV (x) əx 2

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We consider an object of mass m moving in one spatial dimension. We denote its position by r(t) in a given system of coordinates where t represents the time. With the notations used in the course, we write the Lagrangian of the system as 1 L(x,x) = mx² – V(x), - 2 where the potential energy V(r) does not depend explicitly on any variables other than x. In the lectures we have seen that V(x) is defined through F(x) where F(x) is a force acting on the object. i) Write down the Euler-Lagrange equation(s) of the system and show the equivalence with Newton's equation(s). ƏV (x) əx ii) Derive the Hamiltonian of the system given the definition ƏL(x,x) di L(x, i). H(x,x) = x iii) Show that the Hamiltonian represents that total energy of the system. iv) Show that the Hamiltonian is conserved. v) According to Noether's theorem this conservation is due to a symmetry: in other words, due to the invariance of the system under a certain transformation. What is the symmetry in question? Justify with a few sentences or equations. vi) Show explicitly that this invariance implies that the Hamiltonian is conserved. We now assume that this object is thrown from the ground vertically at time t = 0, with initial velocity of 15m.s-¹ (fifteen metres per second). We take the mass to be m = 1 kg. The altitude of the object is denoted by x. We choose the system of coordinates such that x = 0 at t = 0. The Lagrangian of the system is given by 1 L(x,x) = 2m²² ma² - mgx. = 9.81m.s-2 We assume that g is a constant and we take g vii) Justify the expression of the potential energy V(r) mgr in the previous equa- tion. viii) Was there any other possible expression of V(r) for the same problem? ix) Find the explicit form of r(t) and (t). x) Find the values of t where x reaches its minimum and maximum. xi) What is the maximum altitude max reached by the object? xii) Find the explicit time dependence of the kinetic energy, potential energy and total energy. What do you observe about the time dependence? xiii) Plot your findings from the previous question on the same graph, point out and explain where the various components of the energy reach their minimum and maximum. You can use a software or draw them by hand.
Transcribed Image Text:We consider an object of mass m moving in one spatial dimension. We denote its position by r(t) in a given system of coordinates where t represents the time. With the notations used in the course, we write the Lagrangian of the system as 1 L(x,x) = mx² – V(x), - 2 where the potential energy V(r) does not depend explicitly on any variables other than x. In the lectures we have seen that V(x) is defined through F(x) where F(x) is a force acting on the object. i) Write down the Euler-Lagrange equation(s) of the system and show the equivalence with Newton's equation(s). ƏV (x) əx ii) Derive the Hamiltonian of the system given the definition ƏL(x,x) di L(x, i). H(x,x) = x iii) Show that the Hamiltonian represents that total energy of the system. iv) Show that the Hamiltonian is conserved. v) According to Noether's theorem this conservation is due to a symmetry: in other words, due to the invariance of the system under a certain transformation. What is the symmetry in question? Justify with a few sentences or equations. vi) Show explicitly that this invariance implies that the Hamiltonian is conserved. We now assume that this object is thrown from the ground vertically at time t = 0, with initial velocity of 15m.s-¹ (fifteen metres per second). We take the mass to be m = 1 kg. The altitude of the object is denoted by x. We choose the system of coordinates such that x = 0 at t = 0. The Lagrangian of the system is given by 1 L(x,x) = 2m²² ma² - mgx. = 9.81m.s-2 We assume that g is a constant and we take g vii) Justify the expression of the potential energy V(r) mgr in the previous equa- tion. viii) Was there any other possible expression of V(r) for the same problem? ix) Find the explicit form of r(t) and (t). x) Find the values of t where x reaches its minimum and maximum. xi) What is the maximum altitude max reached by the object? xii) Find the explicit time dependence of the kinetic energy, potential energy and total energy. What do you observe about the time dependence? xiii) Plot your findings from the previous question on the same graph, point out and explain where the various components of the energy reach their minimum and maximum. You can use a software or draw them by hand.
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