Geometric Numerical Integration: Overview Hamiltonian Systems, - PowerPoint PPT Presentation

Geometric Numerical Integration: Marc Sarbach Geometric Numerical Integration: Overview Hamiltonian Systems, Symplectic Transformations Hamiltonian Systems Examples Hamilton’s Canonical Marc Sarbach Equations Case of quadratic T Symplectic ETH Z¨ urich Transforma- tions introduction January 9 th , 2006 Symplecticity Geometric Interpreta- tion Poincar´ e’s Theorem Characteristic property of Hamiltonian systems

Overview Geometric Numerical Integration: Marc Sarbach Lagrange’s equations Overview Hamiltonian Systems Examples Hamilton’s Canonical Equations Case of quadratic T Symplectic Transforma- tions introduction Symplecticity Geometric Interpreta- tion Poincar´ e’s Theorem Characteristic property of Hamiltonian systems

Overview Geometric Numerical Integration: Marc Sarbach Lagrange’s equations Overview Hamilton’s canonical equations Hamiltonian Systems Examples Hamilton’s Canonical Equations Case of quadratic T Symplectic Transforma- tions introduction Symplecticity Geometric Interpreta- tion Poincar´ e’s Theorem Characteristic property of Hamiltonian systems

Overview Geometric Numerical Integration: Marc Sarbach Lagrange’s equations Overview Hamilton’s canonical equations Hamiltonian Symplectic Transforms Systems Examples Hamilton’s Canonical Equations Case of quadratic T Symplectic Transforma- tions introduction Symplecticity Geometric Interpreta- tion Poincar´ e’s Theorem Characteristic property of Hamiltonian systems

Overview Geometric Numerical Integration: Marc Sarbach Lagrange’s equations Overview Hamilton’s canonical equations Hamiltonian Symplectic Transforms Systems Geometric Interpretation of Symplecticity for non linear Examples Hamilton’s Canonical mappings Equations Case of quadratic T Symplectic Transforma- tions introduction Symplecticity Geometric Interpreta- tion Poincar´ e’s Theorem Characteristic property of Hamiltonian systems

Overview Geometric Numerical Integration: Marc Sarbach Lagrange’s equations Overview Hamilton’s canonical equations Hamiltonian Symplectic Transforms Systems Geometric Interpretation of Symplecticity for non linear Examples Hamilton’s Canonical mappings Equations Case of quadratic T main result: Poincar´ e’s Theorem Symplectic Transforma- tions introduction Symplecticity Geometric Interpreta- tion Poincar´ e’s Theorem Characteristic property of Hamiltonian systems

Overview Geometric Numerical Integration: Marc Sarbach Lagrange’s equations Overview Hamilton’s canonical equations Hamiltonian Symplectic Transforms Systems Geometric Interpretation of Symplecticity for non linear Examples Hamilton’s Canonical mappings Equations Case of quadratic T main result: Poincar´ e’s Theorem Symplectic Transforma- Preservation of Hamiltonian character under symplectic tions transformations introduction Symplecticity Geometric Interpreta- tion Poincar´ e’s Theorem Characteristic property of Hamiltonian systems

introduction Geometric Numerical Suppose, that the position of a mechanical system with d Integration: degrees of freedom described by Marc Sarbach q = ( q 1 , . . . , q d ) T , Overview Hamiltonian as generalized coordinates, such as cartesian coordinates, Systems angles etc. We suppose, that the kinetic energy is of the Examples Hamilton’s form Canonical Equations T = T ( q, ˙ q ) Case of quadratic T Symplectic and the potential energy is of the form Transforma- tions introduction U = U ( q ) . Symplecticity Geometric Interpreta- tion We then define L = T − U as the corresponding Lagrangian Poincar´ e’s Theorem Characteristic of the system. property of Hamiltonian systems

introduction Geometric Numerical Integration: Marc Sarbach The coordinates q 1 ( t ) , . . . , q d ( t ), then obey the set of Overview differential equations Hamiltonian � ∂L Systems d � − ∂L = 0 , for k = 1 , . . . , d. Examples dt ∂ ˙ q k ∂q k Hamilton’s Canonical Equations Case of Numerical or analytical integration of this system therefore quadratic T Symplectic allows one to predict the motion of the system, given the Transforma- tions initial values. introduction Symplecticity Geometric Interpreta- tion Poincar´ e’s Theorem Characteristic property of Hamiltonian systems

Examples Geometric Numerical Newton’s second law Integration: Let m be a mass point in R 3 with Cartesian coordinates Marc Sarbach ( x 1 , x 2 , x 2 ) T . We have T = 1 x 2 x 2 x 2 2 m ( ˙ 1 + ˙ 2 + ˙ 3 ). Suppose, the Overview point moves in a conservative force field F ( x ) = −∇ U ( x ). Hamiltonian Calculation of the Lagrangian equations leads to Systems m ¨ x − F ( x ) = 0, which is Newton’s second law. Examples Hamilton’s Canonical Equations Case of quadratic T Symplectic Transforma- tions introduction Symplecticity Geometric Interpreta- tion Poincar´ e’s Theorem Characteristic property of Hamiltonian systems

Examples Geometric Numerical Newton’s second law Integration: Let m be a mass point in R 3 with Cartesian coordinates Marc Sarbach ( x 1 , x 2 , x 2 ) T . We have T = 1 x 2 x 2 x 2 2 m ( ˙ 1 + ˙ 2 + ˙ 3 ). Suppose, the Overview point moves in a conservative force field F ( x ) = −∇ U ( x ). Hamiltonian Calculation of the Lagrangian equations leads to Systems m ¨ x − F ( x ) = 0, which is Newton’s second law. Examples Hamilton’s Canonical Equations Case of Pendulum quadratic T Symplectic Take α as the generalized coordinate. Since x = l sin( α ) and Transforma- tions y = − l cos( α ), we find for the kinetic energy introduction x 2 + ˙ 2 ml 2 ˙ α 2 and for the potential energy T = 1 y 2 ) = 1 2 m ( ˙ Symplecticity Geometric Interpreta- U = mgy = − mgl cos( α ). The Lagrangian equations then tion α + g Poincar´ e’s lead to ml 2 ¨ l sin( α ) = 0, the pendulum equation. Theorem Characteristic property of Hamiltonian systems

Hamilton’s Canonical Equations Geometric Numerical Integration: Hamilton simplified the structure of Lagrange’s equations. Marc He introduced the conjugate momenta: Sarbach p k = ∂L Overview for k = 1 , . . . , d (1) ∂ ˙ Hamiltonian q k Systems Examples and defined the Hamiltonian as Hamilton’s Canonical Equations H ( p, q ) := p T ˙ Case of q − L ( q, ˙ q ) , quadratic T Symplectic Transforma- by expressing every ˙ q as a function of p and q , i.e. tions introduction q = ˙ ˙ q ( p, q ). Here it is, required that (1) defines, for every q , Symplecticity Geometric a continuously differentiable bijection: ˙ q ↔ p . This map is Interpreta- tion called Legendre Transformation. Poincar´ e’s Theorem Characteristic property of Hamiltonian systems

Equivalence of Hamilton’s and Lagrange’s equations Geometric Numerical Integration: Marc Sarbach Theorem Overview Lagrange’s equations are equivalent to Hamilton’s equations Hamiltonian Systems p k = − ∂H ˙ ( p, q ) Examples ∂q k Hamilton’s Canonical Equations q k = ∂H Case of ˙ ( p, q ) , quadratic T ∂p k Symplectic Transforma- tions for k = 1 , . . . , d . introduction Symplecticity Geometric Interpreta- tion Poincar´ e’s Theorem Characteristic property of Hamiltonian systems

Case of quadratic T Geometric Numerical q T M ( q ) ˙ Assume T = 1 Integration: 2 ˙ q quadratic, where M ( q ) is a Marc symmetric and positive definite matrix. For a fixed q we Sarbach q by M − 1 ( q ) p in the definition have p = M ( q ) ˙ q . Replacing ˙ Overview of the Hamiltonian leads to Hamiltonian Systems H ( p, q ) = p T M − 1 ( q ) p − L ( q, M − 1 ( q )) Examples Hamilton’s = p T M − 1 ( q ) p − 1 2 p T M − 1 ( q ) p + U ( q ) Canonical Equations Case of quadratic T = 1 2 p T M − 1 ( q ) p + U ( q ) , Symplectic Transforma- tions introduction which is the total energy of the system. For quadratic Symplecticity Geometric Interpreta- kinetic energies, the Hamiltonian therefore represents the tion Poincar´ e’s total energy. Theorem Characteristic property of Hamiltonian systems

introduction Geometric Numerical Integration: A first property of Hamiltonian systems is, that the Marc Hamiltonian is a first integral for Hamilton’s equations. Sarbach Another very important property, which will be shown later, Overview is the symplecticity of its flow. The basic objects we study Hamiltonian are two-dimensional parallelograms in R 2 d . Suppose, that a Systems parallelogram is spanned by two vectors Examples Hamilton’s Canonical Equations � ξ p � η p � � Case of ξ p , ξ q , η p , η q ∈ R d , quadratic T ξ = , η = ξ q η q Symplectic Transforma- tions in the p, q -space. Therefore, the parallelogram is defined as introduction Symplecticity Geometric Interpreta- P := { tξ + sη | 0 ≤ t ≤ 1 , 0 ≤ s ≤ 1 } tion Poincar´ e’s Theorem Characteristic property of Hamiltonian systems