Lagrangian Hamiltonian Dynamics [BETTER]
The best of all is, Lagrangian is a powerful mathematical method tosolve problems in classical mechanic but Hamiltonian is a powerfulmethod to solve problems in classical mechanics, quantum mechanics,statistical mechanics, thermodynamics... etc actually almost all physics...
Lagrangian Hamiltonian dynamics
In planetary dynamics, there is a large separation of scales between the interaction of planets with the central star and their mutual interactions. "Secular theory" describes the very long term evolution of the system using Hamiltonian mechanics. You can apply a canonical transformation (Von Zeipel transformation) along the action-angle variables of the short term interactions. You can then derive the long term evolution, (for example that of the eccentricities and inclinations), investigate whether the long-term perturbing effects of planets add up resonantly or not, whether the system is chaotic, etc.
Where $I_n\times n$ is the identity matrix, with a system of $n$ spatial coordinates (and thus, $n$ speeds, and those, $2n$ amounts of phase space coordinates). Also, for an observable $G$, we have: $\dot G = \G, H\$ as you know. So, you can easily have the dynamics of an given observable $G$. All very nice and neat and general, but....
Where operator $X_G$ is acting on $\xi^i$. We can solve the differential equation in successive infinitesimal transformations, arriving in the fundamental exponential limit, thus having the complete general solution of any hamiltonian system for any observable $G$:$$\xi^i(\epsilon) = \exp\left(\Delta\epsilon X_G\right)\xi^i_0$$
Do you understand the power of it?? Pointing out, again, here it is the solution of any hamiltonian system for any observable $G$ with parameter $\epsilon$ generated by operator $X_G$. If you want to analyse dynamics, then $\epsilon$ is the time, and $G$ is the hamiltonian, where $X_H$ defines the hamiltonian vector space. All hamiltonian systems have the same solution. The same solution!! So, lets solve for the dynamics (ie, where $\epsilon$ is time):$$\xi^i(t) = \exp\left(\Delta t\fracddt\right)\xi^i_0$$
Studying Hamiltonian dynamics/symplectic manifolds becomes particularly useful when space time is not Euclidian. For example symplectic manifolds and hence Hamilton dynamics do not exist on a sphere $S^2n$ for n>1. So it is these types of questions that can very naturally be studied in the symplectic manifold/Hamiltonian setting rather than the Lagrangian formalism.
So the Hamilton equation is important because it makes it possible to treat independent systems independent in the information theoretic (from which follows probabilistic) frameworks, as this was hinted at the first point in the first answer. Statistical mechanics is based on this. Also, thermodynamics would not exist with the concept of Energy. Since independent systems are described by their energy which is extensive, additive.
Interestingly, all extensive variables in thermodynamics are related to changed of the phase space. Volume grows, volume related phase space changes, kinetic energy decreases (momentum related phase space decreases), in adiabatic systems, so that the total information content stays constant (and consequently the entropy).
This book provides an accessible introduction to the variational formulation of Lagrangian and Hamiltonian mechanics, with a novel emphasis on global descriptions of the dynamics, which is a significant conceptual departure from more traditional approaches based on the use of local coordinates on the configuration manifold. In particular, we introduce a general methodology for obtaining globally valid equations of motion on configuration manifolds that are Lie groups, homogeneous spaces, and embedded manifolds, thereby avoiding the difficulties associated with coordinate singularities.
An introductory textbook exploring the subject of Lagrangian and Hamiltonian dynamics, with a relaxed and self-contained setting. Lagrangian and Hamiltonian dynamics is the continuation of Newton's classical physics into new formalisms, each highlighting novel aspects of mechanics that gradually build in complexity to form the basis for almost all of theoretical physics. Lagrangian and Hamiltonian dynamics also acts as a gateway to more abstract concepts routed in differential geometry and field theories and can be used to introduce these subject areas to newcomers.Journeying in a self-contained manner from the very basics, through the fundamentals and onwards to the cutting edge of the subject, along the way the reader is supported by all the necessary background mathematics, fully worked examples, thoughtful and vibrant illustrations as well as an informal narrative and numerous fresh, modern and inter-disciplinary applications.The book contains some unusual topics for a classical mechanics textbook. Most notable examples include the 'classical wavefunction', Koopman-von Neumann theory, classical density functional theories, the 'vakonomic' variational principle for non-holonomic constraints, the Gibbs-Appell equations, classical path integrals, Nambu brackets and the full framing of mechanics in the language of differential geometry.
The Lagrangian is a function with dimensions of energy that summarises the dynamics of a system. The equations of motion can be obtained by substituting into the Euler-Lagrange equation.
This module introduces some fundamental concepts in analytical dynamics and illustrates their applications to relevant problems. The module covers the calculus of variations, Lagrangian and Hamiltonian formulations of dynamics, Poisson brackets, canonical transformations and Hamilton-Jacobi equations.The module leads, among other things, to a deeper understanding of the role of symmetries and conservation laws. This course lays the foundations for the Year 3 module Quantum Mechanics (MAT3039).
Just wondering the distinct differences between lagrangian and hamiltonian mechanics. I'm assuming hamiltonian mechanics is used for small particle systems, either single particle or low order perturbations. And I'm at a loss for the advantage of lagrangian mechanics. I know it has to do with field theory but have yet to fully get into lagrangians and what not.
Lagrangian and Hamiltonian dynamics have inspired several promising optimization and sampling algorithms such as first-order methods in optimization, Hamiltonian Monte Carlo (see also this paper that will appear in NIPS 2018). Legendre duality also appears in convex optimization as Fenchel duality. This note, written primarily for optimization folks, introduces Lagrangian dynamics, Hamiltonian dynamics, and proves the duality that connects them.