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Hamilton’s Principle in Phase Space

This chapter derives Hamilton’s equations using the Legendre transform and the definition of the Hamiltonian function. While, in the Newtonian formalism, conservation laws were rather difficult to tease out, the Lagrangian formalism revolutionised the way of looking at them; however, the Hamiltonian formalism is perhaps even simpler than the Lagrangian formalism, making it straightforward to identify conservation laws and the symmetries of the system associated with each conserved property. In this chapter, the Hamiltonian is treated as being explicitly dependent on time, as this form is more general and will lead to an important relation that, although not an equation of motion, is still useful to discuss. The chapter also introduces Routhian mechanics as a symplectic reduction technique, using integrals of the motion.